Moduli Spaces of Pseudo-Holomorphic Disks and Floer Theory of Cleanly Intersecting Immersed Lagrangians

MODULI SPACES OF PSEUDO-HOLOMORPHIC DISKS AND FLOER THEORY OF CLEANLY INTERSECTING IMMERSED LAGRANGIANS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Ken Yin Kwan Chan August 2010

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/bz202yk0512

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Eleny Ionel, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Yakov Eliashberg

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Soren Galatius

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

In this thesis we investigate moduli spaces of pseudo-holomorphic disks with La- grangian boundary conditions, in which the Lagrangians are immersed with clean self-intersections. We then discuss the compactiﬁcation of these moduli spaces, and show that under speciﬁc assumptions, the moduli spaces can be oriented. Finally, we use these moduli spaces to construct and compute Lagrangian Floer cohomology for sphere and orientation covers of ℝPn embedded as a Lagrangian submanifold of ℂPn.

iv Preface

There have been tremendous interests in pseudo-holomorphic curves with Lagrangian boundary conditions. For instance, they enter in the definition of Floer’s homology theory of Lagrangian intersections, and constitute the major framework of “La- grangian intersection Floer theory – anomaly and obstruction” [FOOO06], in which Fukaya, Oh, Ohta and Ono laid the foundations for constructing the derived Fukaya Category of a symplectic manifold. The later plays an essential role in Kontsevich’s Homological Mirror Symmetry Conjecture. While significant steps were taken to understand the Fukaya Category in Fukaya et. al. [FOOO06] , defining it in complete generality and making rigorous the algebraic structures involved is a long term and technically involved project. Na¨ıvely, the objects in the category are embedded Lagrangians with extra information. Insofar, morphisms between pairs of transversely intersecting Lagrangians are defined. They are induced from the intersections of the Lagrangians, with algebraic operations determined by the information extracted from pseudo-holomorphic discs bounding these Lagrangians. But this only leads to an ‘precategory’ as morphisms are merely defined for pairwise transversely intersecting objects. That said, to obtain a ‘category’, one is led to consider immersed Lagrangians, and allow for intersections other than pairwise transverse ones. Akaho and Joyce took the initial step in [AJ08] to investigate immersed Lagrangians whose self-intersections are transverse double points. This motivates the author to take a further step in extending their work to cover a broader class of immersed Lagrangians, which may possess certain non-transverse self-intersections. It is an on-going project built on the technical machinery of [FOOO06].

v Closely related to the Fukaya Category is the Donaldson-Fukaya Category. It is deﬁned at the (co)-homology level with Floer cohomologies as morphisms, which are associative. This makes it technically more tractable, but at cost of losing information required for Mirror Symmetry after passing to cohomology. Nevertheless, it justiﬁes the attention on Lagrangian Floer cohomology, which can be viewed as one of the by-products of the moduli spaces the author proposes to investigate.

vi Acknowledgment

First and foremost, I would like to thank my thesis advisor, Eleny Ionel. Over the years, she has been extremely generous with her time, and her advice was always remarkably insightful. I am certainly indebted to her for her patience and her constant encouragement. I would also like to thank Yasha Eliashberg for introducing me to the beautiful subject of symplectic topology, and Sam Lisi for many interesting and helpful discussions. I am deeply appreciative of all those who has given me support and encouragement during my time at Stanford.

vii Contents

Abstract iv

Preface v

Acknowledgment vii

1 Introduction 1 1.1 Clean self-intersections ...... 1 1.2 A local model for clean self-intersections ...... 3

2 Moduli spaces 5 2.1 Deﬁnition of the moduli spaces ...... 6 2.1.1 Lagrangian boundary conditions ...... 6

2.1.2 Evaluation maps on ℳm+1(J, ℐ, ) ...... 8 2.2 Compactiﬁcation and boundary ...... 8 2.2.1 Lagrangian boundary conditions for stable maps ...... 9

2.2.2 Evaluation maps on ℳm+1(J, ℐ, ) ...... 12 2.3 The linearization ...... 13 2.4 The virtual dimension ...... 17

3 Orientability of the moduli spaces 25 3.1 Preliminary deﬁnitions ...... 25 3.2 Orientation of the moduli spaces ...... 28

viii 4 Lagrangian Floer cohomology 31 4.1 A brief overview ...... 31

5 Examples and computations 34 5.1 Sphere double covers of ℝPn in ℂPn ...... 34

zk 5.2 The moduli spaces ℳ2 ( ) with one jump ...... 36

5.2.1 Regularity of D∂J0 ...... 37

zk 5.2.2 Orienting the moduli spaces ℳ2 ( j ) ...... 40 5.3 Morse-Bott Floer cohomology of the sphere covers of ℝPn ...... 42 5.3.1 Novikov Ring ...... 43 5.3.2 Floer coboundary operators ...... 44 5.3.3 Spectral sequence and Floer cohomology ...... 47 5.4 Morse-Bott Floer cohomology of the orientation covers of ℝPn . . . . 55

5.4.1 Floer coboundary operators and ℳ2( ) ...... 57

6 Further directions 61 6.1 Obstructions to Lagrangian Floer cohomology ...... 61 6.2 Hamiltonian equivalence ...... 61

A Spin and relatively spin structures 63

References 66

ix Chapter 1

Introduction

The spaces under consideration are a 2n-dimensional symplectic manifold (X,!) with a symplectic structure !, and a smooth n-dimensional manifold L. Throughout this thesis, all manifolds are smooth and without boundary, unless otherwise stated. Sup- pose : L ↬ X is a proper Lagrangian immersion whose image (L) lies in X. By a Lagrangian immersion we mean is a proper immersion with ∗! = 0. The immersion identiﬁes the tangent bundle TL of L with a Lagrangian subbundle of the symplectic bundle (∗T X, ∗!). When (X, J, !) is almost K¨ahlerwith a compatible almost complex structure J, one can show that an immersion : L ↬ X is Lagrangian if and only if ∗J maps TL to the normal bundle NL of L in ∗TX.

1.1 Clean self-intersections

In [BW97], Bates and Weinstein define the clean intersection of two maps to the same target manifold in terms of their fiber product. We adapt their definition to the case of two identical maps: consider the fiber product,

pr2 L×L / L (1.1) FF F prX pr1 FF FF F" L / X

1 CHAPTER 1. INTRODUCTION 2

where pr1 and pr2 are projections on the ﬁrst and on the second factors respectively.

In other words, L×L is the pre-image of the diagonal ΔX :X,→ X ×X under the map pr1×pr2. It follows that the ﬁber product L×L contains a component consisting i of the diagonal ΔL of L . In this and subsequent sections, ℛ will denote the closed ii set (L×L) ∖ ΔL, and K will be the set pr1(ℛ), which is identical to pr2(ℛ) , with an image (K) under . L×L can then be expressed as the disjoint union ΔL∐ℛ.

If the ﬁber product L×L happens to be a smooth submanifold of L×L, then for any tangent vector (vx, wy) in T(x,y)(L×L), d(×)(x,y) (vx, wy) is tangent to the diagonal ΔX in X×X, which implies that dx(vx) = dy(wy). But d(pr1)x (vx, wy) = vx, and d(pr1)x (vx, wy) = 0 only when vx = 0. In this case, dx(vx) = dy(wy) = 0. It then follows that the projection pr1 is an immersion. Similarly, pr2 is also an immersion as well. Therefore, we conclude that the projection

prX : L×L −→ X

is itself an immersion if L×L is a smooth submanifold of L×L, in which case the image of L×L under prX is an immersed submanifold of X. This observation motivates the following deﬁnition of clean self-intersections.

Deﬁnition 1.1. An immersion : L ↬ X is said to intersect itself cleanly provided that the ﬁber product L×L is a smooth submanifold of L×L, and for any point (x, y) in L×L,

dx(TxL) ∩ dy(TyL) = d( ∘ pr1)(x,y)T(x,y) (L×L) (1.2)

∗ as subbundles of ( ∘ pr1) TX.

This requires ℛ to be a smooth submanifold of L×L, and consequently L×L is diffeomorphic to L∐ℛ. In general, the fiber product L×L may consist of components of possibly different dimensions, which by condition (1.2) are bounded above by the dimension of L. In the special case where -1(p), for all points p in X and a cleanly self-intersecting immersion , has at most two distinct points, the set ℛ, considered

i The closeness of (L×L) ∖ ΔL in L×L follows from being an immersion. ii Since there is an involution : L×L → L×L given by (x, y) = (y, x). CHAPTER 1. INTRODUCTION 3

as a relation on K×K, deﬁnes an injection on K, which implies that the projections prj’s are embeddings and K is itself an embedded submanifold of L.

Example 1.2. Consider an embedded, but not necessarily orientable, Lagrangian submanifold (L) of X. Its orientation double cover :O(L) → (L) is then a La- grangian immersion, with ℛ being diﬀeomorphic to O(L). A particularly interesting example is ℝPn ,→ ℂPn for n even with orientation cover Sn → ℝPn given by the antipodal map. ℝPn is not orientable in this case.

Example 1.3. Three lines in the plane intersecting each others in a single point is a cleanly self-intersecting immersed Lagrangian, where L in this case is the disjoint union of the three lines. Another example is when three planes in ℝ4 intersecting each others in a common point, where L is the disjoint union of the planes.

1.2 A local model for clean self-intersections

-1 Suppose p is a point in (K). For any two distinct points x1 and x2 of L in (p), there exist connected and pairwise disjoint neighborhoods Uj of xj such that ∣Uj is a proper injective immersion, and is thus a smooth embedding, for each j. In particular, Lj =

(Uj) are embedded Lagrangian submanifolds in X. Apply Weinstein’s Lagrangian neighborhood theorem to L1 to obtain a neighborhood V of p symplectomorphic to ∗ the cotangent bundle T L1 of L1, with L1 embedded in it as the 0-section. The self-intersection of : L ↬ X is then modelled in this neighborhood by an isotropic ∗ submanifold W of L1, and L2 is diﬀeomorphic to the conormal bundle of W in T L1, denoted by N ∗W . It is then not hard to see that

TL1∣W + TL2∣W = N1W ⊕ TW ⊕ N2W, and so

TL ∣ + TL ∣ TW = 1 W 2 W , ⊥! (TL1∣W + TL2∣W ) where NjW is the normal bundle of W in Lj. The self-intersection is not assumed to be connected, and the dimensions of the components may be diﬀerent. This includes CHAPTER 1. INTRODUCTION 4

the case of embedded Lagrangians and the case of immersed Lagrangians where self- intersections are all transverse double points. The latter was studied in [AJ08]. Chapter 2

Moduli spaces

In this chapter we investigate moduli spaces of isomorphism classes of stable maps from genus 0 prestable bordered Riemann surfaces bounding immersed Lagrangians with clean self-intersections in the sense of Deﬁnition 1.1. Roughly speaking, the smooth components of the moduli spaces consist of pseudo-holomorphic disks which are solutions to the Cauchy-Riemann equation with boundary on (L). However, the associated Cauchy problem is not well-posed as there is no canonical Lagrangian subspace of TpX at any self-intersection point p in (L). So the Lagrangian Grassmanian of the tangent bundle TX of X has no canonical sections along the boundary of such pseudo-holomorphic disks. In the transverse situation as in [AJ08], any non-constant pseudo-holomorphic disk determines a section of the Lagrangian Grassmanian, continuous along its boundary except at points which are mapped to the self-intersection (L), where it is forced to be discontinuous. The boundary of the given map switches from one branch of L to another at these points. The situation becomes more subtle in the case of clean intersections, as disjoint intervals of the boundary may be mapped entirely into (L). Sections of the Lagrangian Grassmanian are no longer determined by the pseudo- holomorphic disks, and they may switch in between the branches of L multiple times along such intervals. To rectify this problem, we prescribe boundary ‘jump’ points at which a section of the Lagrangian Grassmanian switches branches, and also the branches it lies near

5 CHAPTER 2. MODULI SPACES 6

these marked points. This gives a (possibly discontinuous) section of the Lagrangian Grassmanian, similar to the transverse situation studied in Akaho-Joyce [AJ08] . For simplicity, X and L are assumed to be compact here and in subsequent discussions.

2.1 Deﬁnition of the moduli spaces

We will first give a definition of the smooth components of the moduli spaces and make it precise the Lagrangian boundary conditions satisfied by the pseudo-holomorphic disks in our settings.

2.1.1 Lagrangian boundary conditions

Suppose J is an almost complex structure on X. The triple ( D2, ⃗z, u ) is called a marked J-holomorphic disk with boundary over the almost complex manifold (X,J) if u:D2 → (X,J) is a continuous L1,2 map which is pseudo-holomorphic in the interior, 2 and ⃗z = (z0, . . . , zm) are marked points on the boundary of D . The marked points ⃗z divide the boundary ∂D2 into a collection of arcs with disjoint interiors. Each of these arcs will be labeled jk, with the subscripts inherited from that of the endpoints spanning the arc in the order consistent with the induced orientation on the boundary. Choose an almost complex structure J compatible with !, and consider the almost K¨ahlermanifold (X,!,J).

Deﬁnition 2.1. Let ℐ be a ﬁxed subset of {0, . . . , m}. A marked J-holomorphic disk ( D2, ⃗z, u ) with boundary is said to satisfy the Lagrangian boundary condition of type ℐ if there exist a set continuous maps uejk : jk → L for all the arcs in the collection such that

1. ∘ u = u ; ejk ∣ jk

2. (ue★j(zj), uej⋄(zj)) lies in ℛ, whenever j belongs to ℐ. In this case, we call zj a jump point;

3. (ue★j(zj), uej⋄(zj)) lies in ΔL, if j does not belong to ℐ. CHAPTER 2. MODULI SPACES 7

The collection ue = {ue★⋄} is called a lift of u subordinate to the index set ℐ. It induces a map 2 ue ∂D −−−→ L×L ⏐ ⏐ ⏐ ⏐pr y y X (D2, ∂D2) −−−→ (X, (L)) u which will also be denoted synonymously as ue. By deﬁnition, ue is not continuous at the jumps. In other words, at each jump zj,

u(zj) = lim u∣∂Σ(s), lim u∣∂Σ(s) ∈ ℛ. e s↗zj s↘zj

2 2 2 Any biholomorphic map ' : D → D acts on the tuple ( D , ⃗z, u, ue ) by

2 2 −1 ' ⋅ ( D , ⃗z, u, ue ) = ( D ,' (⃗z) , u ∘ ', ue ∘ ' ) ,

2 2 and this defines an isomorphism between the tuples ( D , ⃗z, u, ue ) and ' ⋅ ( D , ⃗z, u, ue ). 2 ' defines an automorphism of ( D , ⃗z, u, ue ) if it fixes the tuple.

2 Deﬁnition 2.2. ( D , ⃗z, u, ue ) is called stable if its automorphism group is ﬁnite.

We deﬁne the smooth component ℳm+1(J, ℐ, ) of our moduli space.

Deﬁnition 2.3. Let be a homology class in H2(X, (L); ℤ). Then the moduli space

ℳm+1(J, ℐ, ) is deﬁned as the set of isomorphism classes of such stable tuples

2 2 ℳm+1(J, ℐ, ) = ( D , ⃗z, u, ue ) ∣ u∗[D ] = , tuple is stable / ∼ .

2 2 ′ ′ ′ Here two tuples ( D , ⃗z, u, ue ) ∼ ( D , ⃗z , u , ue ) if and only if there is a biholo- 2 2 2 ′ ′ ′ 2 morphism ' : D → D such that ( D , ⃗z , u , ue ) = ' ⋅ ( D , ⃗z, u, ue ). Denote such 2 isomorphism classes as [ D , ⃗z, u, ue ]. CHAPTER 2. MODULI SPACES 8

2.1.2 Evaluation maps on ℳm+1(J, ℐ, )

There are the usual evaluation maps

evj : ℳm+1(J, ℐ, ) −→ (L)

2 sending a curve ( D , ⃗z, u, ue ) in the moduli space ℳm+1(J, ℐ, ) to the image u(zj) of u at the marked point zj in (L). If the index j happens to be in ℐ, then evj evaluates to the subset (K) of (L). In addition, there are evaluation maps induced by lifts of pseudo-holomorphic maps:

evlj : ℳm+1(J, ℐ, ) −→ L×L

2 which evaluate a curve ( D , ⃗z, u, ue ) by its lift ue at the marked point zj to give a point ue(zj) = (ue★j(zj), uej⋄(zj)) in the ﬁber product L×L, if zj has incident arcs ★j and

j⋄. By deﬁnition, evlj takes its value in ℛ if the index j lies in ℐ. Composing with the projections on the ﬁrst or the second factor of L×L, one obtains a collection of evaluation maps to L:

∓ evlj : ℳm+1(J, ℐ, ) −→ L

− + where evlj stands for the composition pr1 ∘ evlj and evlj is the map pr2 ∘ evlj. Again, for any j in ℐ, these evaluation maps evaluate to the possibly immersed submanifold K of L.

2.2 Compactiﬁcation and boundary

In this section we will show that the moduli space ℳm+1(J, ℐ, ) admits a compactification. It can be compactified by augmenting to it boundary strata consisting of stable pseudo-holomorphic maps defined on genus 0 prestable bordered Riemann surfaces. These stable maps satisfy a similar set of Lagrangian boundary conditions as given in Definition 2.1. CHAPTER 2. MODULI SPACES 9

2.2.1 Lagrangian boundary conditions for stable maps

Suppose J is an almost complex structure on X. The triple ( Σ, ⃗z, u ) is called a marked J-holomorphic curve with boundary over the almost complex manifold (X,J) if Σ is a genus 0 nodal curve with boundary, u : Σ → (X,J) is a continuous L1,2 map c c which is pseudo-holomorphic in the interior Σ of Σ, and ⃗z = (. . . , z0, . . . , zmc ,...) are marked points on the boundary of Σ, where the superscripts c’s index the components of ∂Σ, and these will often be omitted if there are no risks of confusion. The marked points ⃗z, together with the nodal points, divide the boundary of Σ into a collection of arcs with disjoint interiors. Each of these arcs will be labeled

jk as in Section 2.1.1. A subscript a or b will be used instead if the corresponding endpoint is a nodal point. Clearly each nodal point is an endpoint of four such arcs. Choose an almost complex structure J compatible with !, and consider the almost K¨ahlermanifold (X,!,J).

Deﬁnition 2.4. Fix a subset ℐ of {0, . . . , m}. A marked J-holomorphic curve ( Σ, ⃗z, u ) with boundary is said to satisfy the Lagrangian boundary condition of type

ℐ if there exist a set continuous maps uejk : jk → L for all the arcs in the collection such that

1. ∘ u = u ; ejk ∣ jk

2. (ue★j(zj), uej⋄(zj)) lies in ℛ, whenever j belongs to ℐ. Again in this case, we call zj a jump point;

3. (ue★j(zj), uej⋄(zj)) lies in ΔL, if j does not belong to ℐ;

4. (nodal matching) suppose a is a boundary node, with incident arcs ★a and a⋄

from one component of the normalization of ∂Σ, and ♭a and a♯ from the other. Then both of the pairs (ue★a(a), uea♯(a)) and (ue♭a(a), uea⋄(a)) are required to be in ΔL. CHAPTER 2. MODULI SPACES 10

Again, the collection ue = {ue★⋄} is the lift of u. It induces a map

ue ∂Σ −−−→ L×L ⏐ ⏐ ⏐ ⏐pr y y X (Σ, ∂Σ) −−−→ (X, (L)) u which is not continuous at the jumps: at each jump zj,

u(zj) = lim u∣∂Σ(s), lim u∣∂Σ(s) ∈ ℛ. e s↗zj s↘zj

The main difference in Definition 2.4 with the previous one is the last nodal- matching condition. It says lifts in a neighborhood of a boundary node must be consistent with the lifts along the two boundary components of the neck just before it pinches to form that node. Because each boundary component of the neck can only lie in a single branch of the immersed Lagrangian, a node is not regarded as a jump, and thus ue map a node to ΔL. A key ingredient of the aforementioned compactification is Gromov’s compactness.

Theorem 2.5. Let (X2n,!) be a compact symplectic manifold, and :Ln → X2n be a compact Lagrangian immersion with clean self-intersections. Fix an index set ℐ which contains the indices of all the jump points, and let Jn be a sequence of !-compatible almost complex structure in X that converges to a !-compatible almost complex struc- ∞ 2 ture J in the C -topology. Suppose ( D , ⃗zn, un, uen ) is a sequence of Jn-holomorphic 1,p disks of class W , p > 2, with uniformly bounded energy so that supnE(un) < ∞, 2 that satisfy Lagrangian boundary conditions of type ℐ. Then ( D , ⃗zn, un, uen ) has a Gromov convergent subsequence converges to a stable curve (possibly after reparameterizations).

2 Proof. The fact that the sequence ( D , ⃗zn, un ) of Jn-holomorphic maps Gromov converges to some stable map ( Σ, ⃗z, u ) is well-known and its proof can be found, for example, in [MS04]. It remains to show that ( Σ, ⃗z, u ) has a well-deﬁned lift in the sense of Deﬁnition 2.4 when boundary bubblings occur. CHAPTER 2. MODULI SPACES 11

2 First of all, if z ∈ ∂D ∖ {jumps} then uen(z) ∈ ΔL for all n. But since ℛ is closed (this follows from the fact that is an immersion), and disjoint from the diagonal ΔL by deﬁnition, it follows that the limit of (uen(z))n lies in ΔL. Therefore, z remains as a non-jump point in the limit. Next we need to analyze are the jump points and the nodal points in the limit.

Let zj be a jump point. Then uen(zj) is well-defined by definition and takes its value in ℛ. It is not always true that the sequence (uen(zj))n is Cauchy. But the pre-image under of any point in X contains at most finitely many points, and its cardinality is uniformly bounded by the number of components of ℛ, which is finite. Therefore, the sequence (uen(zj))n contains a Cauchy subsequence, and we can always pass to the corresponding subsequence. Hence a lift is well-defined at every jump point.

Let z∞ be a point where the energy accumulates, and let U be a collar neigh- 2 borhood of thickness of a small arc-neighborhood of z∞ on ∂D . By passing to a subsequence and rescaling if necessary, we may assume that z∞ is the only point where energy concentrates on U. Moreover, let z± be two (isolated) points on ∂U which converge to a nodal point in the limit (i.e. a small arc in D2 joining them will have its image pinched to a nodal in the limit). Further, let U,± be neighborhoods of z± in U.

The restrictions vn := un∣V are Jn-holomorphic, and by definition, there is a well-defined lift, continuous away from jump points, ven := uen∣V on ∂U for every n. Because non-constant bubbling requires a minimum amount of energy, by passing to a subsequence if necessary (if jumps are present, we may need to pass further to another ∞ subsequence), vn C -converges to finite bubble trees on every compact subset away from the nodes. In particular, the number of nodes is finite, and clearly lifts in the sense of Definition 2.4 are well-defined in the neighborhood of every node.

Corollary 2.6. Fix X, L and ℐ as in Theorem 2.5. Let Jn be a sequence of !- compatible almost complex structure in X that converges to a !-compatible almost ∞ complex structure J in the C -topology. Suppose ( Σ, ⃗zn, un, uen ) is a sequence of 1,p stable Jn-holomorphic curve of class W , p > 2, with uniformly bounded energy so that supnE(un) < ∞, that satisfy Lagrangian boundary conditions of type ℐ. Then CHAPTER 2. MODULI SPACES 12

( Σ, ⃗zn, un, uen ) has a Gromov convergent subsequence which converges to a stable nodal curve.

Deﬁne the moduli space

⎧ 2 ⎫  u∗[D ] = ,  ⎨ ⎬ . ℳ (J, ℐ, ) = ∼ . m+1 ( Σ, ⃗z, u, ue ) ( Σ, ⃗z ) is a genus 0 prestable curve ,   ⎩ ( Σ, ⃗z, u, ue ) is stable ⎭

Corollary 2.7. The moduli space ℳm+1(J, ℐ, ) is compact.

2.2.2 Evaluation maps on ℳm+1(J, ℐ, )

It follows that the evaluation maps in Section 2.1.2 extends to compactiﬁed moduli spaces ℳm+1(J, ℐ, ). We obtain the usual evaluation maps

evj : ℳm+1(J, ℐ, ) −→ (L) sending [ Σ, ⃗z, u, ue ] to u(zj) of u at the marked point zj in (L). Again, for any j ∈ ℐ, evj evaluates to the subset (K) of (L). Similarly, there are evaluation maps induced by lifts:

evlj : ℳm+1(J, ℐ, ) −→ L×L which evaluate [ Σ, ⃗z, u, ue ] by its lift ue at the marked point zj to give a point ue(zj) = (ue★j(zj), uej⋄(zj)) in the ﬁber product L×L, if zj has incident arcs ★j and j⋄. evlj takes its value in ℛ if j ∈ ℐ. Finally, one obtains a collection of evaluation maps to L as in (2.1.2):

∓ evlj : ℳm+1(J, ℐ, ) −→ L

Again, for any j in ℐ, these evaluation maps evaluate to the immersed submanifold K of L. CHAPTER 2. MODULI SPACES 13

2.3 The linearization

In this section, we discuss the linearization of the Cauchy-Riemann problem with immersed Lagrangian boundary conditions. For disjoint union of embedded Lagrangian submanifolds with pairwise transverse intersections, this was done in Fukaya-Oh- Ohta-Ono [FOOO06, Prop. 12.59] and also in Fukaya [Fuk02, Th. 3.2]. Similar con- siderations for the case of a immersed Lagrangian submanifold with transverse double self-intersections was performed in Akaho–Joyce [AJ08, §4.3]. We follow the methods in [FOOO06] and [AJ08] with slight modiﬁcations for our moduli space with lifts.

Fix a point (Σ, ⃗z, j) in the moduli space ℳ0,m+1, where j is some fixed complex structure on the disk Σ and ⃗z is a set of m + 1 distinct marked points {z0 . . . zm} on its boundary. Let ℐ be a finite subset of {0, . . . , m}. By theorem 10.4 in [FO97] (cf. [Fuk02]), one can find a one dimensional Kählermanifold Δ such that:

1. there exists a compact subset Δc with Δ ∖ Δc being isometric to ∣ℐ∣ disjoint

copies of (−∞, 0]×[0, 1], each of which corresponds to a marked point zk on ∂Σ such that k ∈ ℐ;

2. Δ is conformally equivalent to Σ ∖ {zk ∣ k ∈ ℐ}.

∼ Denote the m + 1 inﬁnite ends by Θk = (−∞, 0]×[0, 1] with coordinates (, t). One can consider them as polar coordinates at the marked points zk’s indexed by ℐ.

Let :Δ → Σ∖{zk ∣ k ∈ ℐ} be the conformal isomorphism. Given any equivalent class [ Σ, ⃗z, u, ue ] in ℳm+1(J, ℐ, ), a smooth map u : (Σ, ∂Σ) → (X, (L)) representing this class induces a map v = u∘ : (Δ, ∂Δ) → (X, (L)). This map has a well-deﬁned lift given by ve = ue∘ ∣∂Δ. Here the normalization map can be taken to be the identity map. Finally, the induced complex structure ∗j on Δ will be denoted as j.

Deﬁnition 2.8. Let ⃗x = (xk ∈ K ⊆ L ∣ k ∈ ℐ ), so that (xk) ∈ (K) for all k ∈ ℐ. 1,p Suppose p > 2 and > 0 is small. Then W (X, (L), ⃗x ) is deﬁned as the set of pairs CHAPTER 2. MODULI SPACES 14

(Δ, v) satisfying the followings:

1,p v : (Δ, ∂Δ) → (X, (L)) is of class Wloc ; (2.1) Z ∣∣′ p e { ∣dv∣ + (v(, t), (xk)) } ddt < ∞ for any k, (2.2) Θk where is the Riemannian distance on X induce by the !-compatible almost complex ′ ′ ′ structure J, and ∣⋅∣ is a smooth function such that ∣∣ = ∣∣ for ∣∣ ⩾ 1 and ∣∣ = 0 for ∣∣ ⩽ 1/2.

1,p Standard analysis shows that the space W (X, (L), ⃗x ) is Banach. The condition in (2.2) is equivalent to requiring v∣Θk (, t) to converge to some point in the self- intersection (K) of : L ↬ M as → −∞ uniformly in t.

Lemma 2.9. For any pair (Δ, v) in deﬁnition 2.8, the map v ∘ −1 extends contin- uously to a function v on Σ for any p > 2.

1,p Deﬁnition 2.10. Denote by Wf (X, (L), ⃗x ) the set of tuples (Δ, v, ve) satisfying the following properties:

1,p 1. (Δ, v) is an element in W (X, (L), ⃗x ),

2. there exists a continuous map ve : ∂Δ → L such that for all k ∈ ℐ,

lim v∣Θ (, 0) = xk, →−∞ e k

3. lim v∣Θ (, 0), lim v∣Θ (, 1) ∈ ℛ for all k ∈ ℐ. →−∞ e k →−∞ e k

It follows that ve induces a lift of the map v :(Σ, ∂Σ) → (X, (L)). Notice that 1,p when the space Wf (X, (L), ⃗x ) is non-empty, it forms a ﬁnite cover of the space 1,p W (X, (L), ⃗x ) for any ⃗x ﬁxed. 1,p 1,p ∣ℐ∣ Let Wf (X, (L), ℐ) be the union of Wf (X, (L), ⃗x ) as ⃗x ranges over (pr1ℛ) = K∣ℐ∣. There is a projection

1,p ∣ℐ∣ pr : Wf (X, (L), ℐ) −→ ℛ CHAPTER 2. MODULI SPACES 15

given by pr (Δ, v, v) = lim v∣Θ (, 0), lim v∣Θ (, 1) , k e →−∞ e k →−∞ e k

1,p whose ﬁber is Wf (X, (L), ⃗x ). Consider the Banach space bundle

1,p ℰ −→ Wf (X, (L), ℐ)

1,p whose ﬁber at v ∈ Wf (X, (L), ℐ) is the Banach space

p 0,1 ∗ ℰv := L Δ, Ω (T Δ) ⊗ E ,

∗ where E is the bundle v (TX) and v = (Δ, v, ve) . Then for any ﬁxed !-compatible almost complex structure J, there is a section

1,p n 2m+1 o ∂J : Wf (X, (L), ℐ)× ∂D ∖ Δ −→ ℰ, (2.3) where m + 1 is the number of marked points and Δ is the subest of (∂D2)m+1 in which two marked points coincide, given by the Cauchy-Riemann operator,

1 ∂ (v) := (dv + J ∘ dv ∘ j) . J 2

The moduli ℳm+1(J, ℐ, ) can then be identiﬁed with the subset

−1 1,p ∂J (0) ⊆ Wf (X, (L), ℐ) of elements satisfying the Cauchy-Riemann equation, modulo reparameterizations by biholomorphisms of the domain. 1,p Given any (Δ, v, ve) in Wf (X, (L), ℐ), the variations in the vertical direction along the ﬁber, while keeping pr(Δ, v, ve) ∈ ℛ ﬁxed, is given by the following space.

1,p Deﬁnition 2.11. Suppose > 0 is small, and (Δ, v, ve) is in Wf (X, (L), ℐ). Fix a ∗ 1,p ∗ ∗ Levi-Civita connection ∇ of the bundle v TX. Then deﬁne W (Δ, v TX; ve TL) as CHAPTER 2. MODULI SPACES 16

the set of sections of v∗TX which satisfy:

∈ W 1,p(Δ, v∗TX), such that (x) ∈ T L for all x ∈ ∂Δ; (2.4) loc ve(x)

Z ′ e∣∣ (∣∣p + ∣∇∣p) ddt < ∞ for any k, (2.5) Θk

′ ′ ′ where ∣⋅∣ is a smooth function such that ∣∣ = ∣∣ for ∣∣ ⩾ 1 and ∣∣ = 0 for ∣∣ ⩽ 1/2.

The linearization of ∂J at a pseudo-holomorphic map v in (Δ, v, ve) , while keeping the domain and the base point in ℛ ﬁxed, gives a Fredholm operator

1,p ∗ ∗ p 0,1 ∗ ∗ D := Dv∂,J : W (Δ, v TX; ve TL) → L(Δ, Ω (T Δ) ⊗ v TX) (2.6) between the Banach spaces.

1,p Lemma 2.12. Given any (Δ, v, ve) ∈ Wf (X, (L), ℐ) with the map

v : (Δ, ∂Δ) −→ (X, (L))

∗ and the associated lift ve, the bundle ve TL is isomorphic to a Lagrangian subbundle ∗ of (v∣∂Δ) TX.

Proof. The Lagrangian immersion : L ↬ M induces an injective bundle homomor- ∗ ∗ ∗ ∗ phism d : TL → TX. By deﬁnition v∣∂Δ = ∘ ve, and so (v∣∂Δ) TX = ve TX. ∗ ∗ There is a bundle map ∗ : ve TL → (v∣∂Δ) TX induced by d which we describe as ∗ ∗ follows: since ∂Δ is a disjoint union of copies of ℝ, the bundle ve TL and (v∣∂Δ) TX are both trivial. Choose any trivializations of the bundles. The map ∗ is then

(x, ) = d ( ), ∗ x ve(x) x for any x ∈ ∂Δ and ∈ T L. The injectivity of follows from the injectivity of x ve(x) ∗ ∗ d, and the image of ∗ is a Lagrangian subbundle of (v∣∂Δ) TX. CHAPTER 2. MODULI SPACES 17

∗ Lemma 2.12 allow us to identify ve TL with its image under the map ∗ to obtain a bundle pair ∗ ∗ (v TX, ve TL) → (Δ, ∂Δ). (2.7) A section of the bundle pair (2.7) is a section of v∗TX such that (x) ∈ T L for all ve(x) −1 ∗ ∗ x ∈ ∂Δ (here we identify (x) with (∗) (x) for x ∈ ∂Δ). Hence (v TX, ve TL) gives a well-deﬁned Lagrangian boundary condition for the Fredholm problem in (2.6).

2.4 The virtual dimension

We will calculate the virtual dimension of the moduli space ℳm+1(J, ℐ, ) in this section. The operator in (2.6) is of product type at the ends Θk. More precisely, let us consider the canonical embedding ek :Θk ,→ Δ. The pull-back of the operator

Dv∂J by ek to Θk has the form ∂ + A . As → ∞, the operator A converges to the operator Jpk ∂t, which is self-adjoint. So after a compact perturbation, we may assume there exists a negative number M such that on the subset Θk,M := (−∞,M)×[0, 1], the operator Dv∂J has the form ∂ + A (2.8) ∂ where A = J0∂t. Denote the operator in (2.8) by ∂0. The operator A is related to the self–intersection (K) according to the following observation by Floer in [Flo88b, §2].

Lemma 2.13 ([Flo88b, §2]). Given any point p in the self–intersection (K), we have ∼ ker Ak = Tp(K), where Ak is deﬁned as in (2.8) corresponding to Θk.

For each k = 1,..., ∣ℐ∣, consider the restrictions of the lift ve on Θk,M . Let

p− = lim v(, 0), and p+ = lim v(, 1). k →−∞e k →−∞e

± ± There exists a neighborhood Vk of pk in X, and neighborhoods Uk of pk in L such ± ± ± that ± := ∣ ± : U → V are smooth embeddings. Further let L = U be the Uk k k k corresponding embedded Lagrangian neighborhoods. By choosing Vk small enough, −1 we may assume that v (V ) contains Θk,M . With this local models, by restricting to CHAPTER 2. MODULI SPACES 18

Θk,M , the bundle pair in (2.7) can then be trivialized as

± Θk,M ×Tp X, ∂ Θk,M ×T ± L k pk k which gives a well-defined Lagrangian boundary condition for (2.8) at Θk,M . We are now going to cap off the jumps at each end. Consider the set C defined as

{z ∈ ℂ ∣ ∣z∣ ⩽ 1} ∪ {z ∈ ℂ ∣ Re z ⩾ 0, ∣Im z∣ ⩽ 1} .

± ± Let Λ = d T ± L be Lagrangian subspaces of Tp X. We choose a smooth path k pk k k ± in the Lagrangian Grassmanian Lag(Tpk X) joining Λk along the boundary of C as follows: let

k : ∂ C −→ Lag(Tpk X) be a map such that for any point (, t) = + it ∈ ∂ C,

⎧ − ⎨Λk if t = 1, k(, t) = (2.9) + ⎩Λk if t = −1.

Identify ∂ C with ℝ by a diﬀeomorphism : ℝ → ∂ C such that ⎧  − i if s ⩾ 1, ⎨ i( 1 −s) (s) = e 2 if 0 ⩽ s ⩽ 1,   ⎩ + i if s ⩽ 0.

′ Composing with the map , k = k ∘ can be considered as a map from the interval

[0, 1] to Lag(Tpk X), and therefore deﬁnes a smooth path in Lag(Tpk X) starting at − + ′ Λk and terminating at Λk . Finally, deﬁne the path k by

′ ′ k(s) = k(1 − s).

This corresponds to the map k given by reﬂecting k about the –axis: k(, t) = ′ k(, −t). For simplicity of notations, in the followings we will identify k with k CHAPTER 2. MODULI SPACES 19

and refer it as a path in the Lagrangian Grassmanian.

Deﬁnition 2.14. Let > 0 be small, and vk : C → X be the constant map mapping 1,p C to the point pk in (K). Deﬁne W (C,Tpk X; k) as the set of sections of the ∗ ∼ trivial bundle vkTX = C×Tpk X → C such that

1,p is of class Wloc , (2.10)

∣∂ C(, t) ∈ k(, t) for all (, t) ∈ ∂ C, (2.11) Z e∣∣ (∣∣p + ∣∇∣p) < ∞. (2.12) C

The discussion in (2.8) then leads us to consider the following Cauchy-Riemann operator

1,p p 0,1 ∗ ∂,k := ∂ + Jpk ∂t : W (C,Tpk X; k) → L(C, Ω (T C) ⊗ Tpk X) (2.13) which is Fredholm. The operator in (2.13) consists of the highest order terms of the linearized Cauchy-Riemann operator Dvk ∂J at vk. Denote the Fredholm index of

(2.13) by Ind ∂,k .

Lemma 2.15. Suppose the dimension of the component of (K) where the point pk lies is dpk , then Ind ∂ + Ind ∂ = n − d . ,k ,k pk

Proof. By the homotopy invariance of index, the Fredholm index is invariant under continuous deformations of boundary conditions and coeﬃcients. One may homotope ′ the path k in its own homotopy class to a representative satisfying

′ − − + k(s) ∩ Λk = Λk ∩ Λk = Tpk (K),

for 0 ⩽ s < 1, without altering the index Ind ∂,k . Therefore as in [FOOO06, §12], it − + n n n−dpk dpk suﬃces to assume that Tpk X = ℂ ,Λk = ℝ ,Λk = iℝ ⊕ℝ with the boundary CHAPTER 2. MODULI SPACES 20

n condition given by k : [0, ∞)×[0, 1] → ℂ where

⎧ n ℝ if t = 1, ⎨ n−d d k(, t) = iℝ pk ⊕ ℝ pk if t = 0, (2.14)   n−dp 1  L k ( 2 +aj )i(1−t) dp ⎩ j=1 e ℝ ⊕ ℝ k if = 0.

The problem of calculating the index can be separated into two types of problems.

The ﬁrst type consists of n − dpk problems of ﬁnding sections : [0, ∞)×[0, 1] → ℂ satisfying the equation ∂,k () = 0 and the following boundary conditions:

(, 1) ∈ ℝ, (, 0) ∈ iℝ, (2.15) 1 ( +aj )i(1−t) (0, t) ∈ e 2 ℝ.

The second type consists of dpk problems of ﬁnding holomorphic sections such that

∣∂([0,∞)×[0,1]) ⊆ ℝ, and hence they satisfy the boundary from constant Lagrangian paths associated to the intersection ℝdk . By applying Fourier analysis to solutions satisfying (2.15) as in [RS93, FOOO06], one can deduce that the index of the problems in (2.15) are aj + 1. On the other hand, by similar analysis, solutions to the second type are constants, which are ruled out by the exponential decay condition imposed by the weight factor e∣∣ in (2.12). n−d P pk It follows that the Fredholm index of (2.14) is j=1 (aj + 1).

On the other hand, recall that k is deﬁned as k(, t) = k(, −t) and hence

⎧ + ⎨Λk if t = 1, k(, t) = (2.16) − ⎩Λk if t = −1. CHAPTER 2. MODULI SPACES 21

Similar as before, it suﬃces to investigate the index problem with the following boundary conditions.

⎧ n−dp dp iℝ k ⊕ ℝ k if t = 1, ⎨ n k(, t) = ℝ if t = 0, (2.17)   n−dp 1  L k ( 2 +aj )it dp ⎩ j=1 e ℝ ⊕ ℝ k if = 0.

Again, we are led to consider two types of problems. The ﬁrst requires ﬁnding holomorphic sections : [0, ∞)×[0, 1] → ℂ satisfying the following boundary conditions:

(, 1) ∈ iℝ, (, 0) ∈ ℝ, (2.18) 1 ( +aj )it (0, t) ∈ e 2 ℝ.

The second type is about ﬁnding holomorphic sections such that ∣∂([0,∞)×[0,1]) ⊆ ℝ.

It follows, by the same reasoning as the case for k, that the index corresponding to the ﬁrst type of problem is −aj and the contributions from the solutions to the second n−d type are excluded. Hence the Fredholm index of ∂ is − P pk a . Therefore, ,k j=1 j

n−dp n−dp Xk Xk Ind ∂ + Ind ∂ = a + (n − d ) − a , ,k ,k j pk j j=1 j=1 and the claim of the lemma follows.

We can glue the operator in (2.6) with the operators in (2.13). Their indices then add up to the index of the glued-up operator (see [APS75, LM85]). In the process, one pastes ∣ℐ∣ copies of C to the inﬁnite ends Θk,M of Δ. The boundary of the resulting surface is obtained by capping the ‘openings’ of ∂Δ with ∣ℐ∣ copies of ∂ C. Topologically, this is equivalent to compactifying the domain Δ by adding ∣ℐ∣ points, and by gluing constant maps vk’s to the map v. If Δ and v are induced from a representative (Σ, x; u, ue) of a point in ℳm+1(J, ℐ, ), then compactifying Δ and v CHAPTER 2. MODULI SPACES 22

as described above will give back the triple (Σ, x; u) without altering the homology class. Denote the compactiﬁed pair (Δ, v) as (Δ, v).

The matching conditions (2.9) allow us to paste the lift ve of v with the k’s at ∗ the inﬁnite ends Θk,M to obtain a map ev : ∂Δ → v TL. This permits us to glue the ∗ ∗ bundle pair (v TX, ve TL) in (2.7) with the local models in deﬁnition 2.14 to obtain a bundle pair ∗ ∗ (v TX, ev TL) → (Δ, ∂Δ). (2.19)

There is a well-deﬁned notion of Maslov index for the pair in (2.19) [MS04, RS93], which will be denoted as (d), for a map v with homology class d.

Lemma 2.16. The Fredholm index of the operator in (2.6) is given by

X Ind Dv∂J = n + (d) − Ind ∂,k + dpk . k∈ℐ

Proof. First notice that the operator Dv∂J corresponds to the problem where the map v converges to points pk’s in (K) at the inﬁnite ends Θk, with perturbations which leave pk’s ﬁxed. Similarly, the index

X Ind Dv∂J + Ind ∂,k k∈ℐ corresponds to the problem where jumps in Lagrangian spaces across the punctures of Δ are capped oﬀ by constants maps as in deﬁnition 2.14, but again with points pk’s

ﬁxed. Every point pk can vary in some dk dimensional submanifolds. After adding these contributions, it follows that

X Ind Dv∂J + Ind ∂,k + dpk k∈ℐ gives the index of the problem corresponding to condition (2.19). Hence the lemma follows.

Lemma 2.17. Ind Dv∂J is independent of the choices of k’s. CHAPTER 2. MODULI SPACES 23

Proof. Since the contributions from the Maslov indices in (d) and in

X − Ind ∂,k k∈ℐ cancel each others.

Corollary 2.18. The virtual dimension of ℳm+1(J, ℐ, ) is given by

X vir dimℝ ℳm+1(J, ℐ, ) = n + (d) − Ind ∂,k + (m + 1) − 3. k∈ℐ

Proof. In light of the ﬁbration

1,p 1,p ∣ℐ∣ Wf (X, (L), ⃗x ) −→ Wf (X, (L), ℐ) −→ Im pr ⊆ ℛ , (2.20) the virtual dimension of the moduli space of (parametrized) curves is given by the virtual dimension of the ﬁber in (2.20) and the dimension of the tangent spaces of ∣ℐ∣ Im pr, which can be identiﬁed with the tangent spaces of K (since pr1 : ℛ → K ⊆ L P is an immersion). The latter is given by the sum k∈ℐ dpk , while the former is computed by X Ind Dv∂J = n + (d) − Ind ∂,k + dpk . k∈ℐ Summing up these two contributions, we obtain

X n + (d) − Ind ∂,k . k∈ℐ

Finally, by adding the contributions of varying domain marked points, and subtracting the eﬀect of reparameterizations, we have

X vir dimℝ ℳm+1(J, ℐ, ) = n + (d) − Ind ∂,k + (m + 1) − 3. k∈ℐ

Hence it follows that CHAPTER 2. MODULI SPACES 24

Corollary 2.19. The virtual dimension of ℳm+1(J, ℐ, ) is given by

X vir dimℝ ℳm+1(J, ℐ, ) = n + (d) − k + (m + 1) − 3, k∈ℐ where k is the relative Maslov index of the path k with respect to a based path.

Finally, we remark by Lemma 2.17, the term

X (d) − k k∈ℐ is independent of the paths k. Chapter 3

Orientability of the moduli spaces

In this section we investigate the issue of orientation and the choices required to orient our moduli spaces. This orientation is given by the orientation of the determinant line bundle of an appropriate family of Fredholm operators associated to the moduli spaces.

3.1 Preliminary deﬁnitions

Before we can state Proposition 3.4, we need to introduce certain deﬁnitions and notations. For the rest of the discussions, the Lagrangian L is assumed to be relatively spin and the set ℛ which contains the information of self-intersections is orientable. For any point ⃗p = (x, y) in ℛ whose image under is p, let

ori Ω⃗p := Ω(Lag (TpX); ℒx, ℒy) (3.1) be the space of paths of oriented Lagrangian subspaces of the symplectic vector space

TpX, originating at ℒx := dx(TxL) and terminating at ℒy := dy(TyL). The local orientations of ℒx and ℒy as Lagrangian subspaces of TpX are taken to be the one induced by the respective (oriented) tangent spaces of L at the points x and y. The

ori space Lag (TpX) in (3.1) is modelled on the homogeneous space U(n)/SO(n). In general, the components of Ω⃗p are separated by the relative Maslov index of the

25 CHAPTER 3. ORIENTABILITY OF THE MODULI SPACES 26

oriented Lagrangian paths: there is an isomorphism

=∼ 0,⃗p : 0(Ω⃗p) −→ ℤ

given by the relative Maslov index 0,⃗p with respect to a chosen path 0,⃗p in Ω⃗p.

Denote the union of Ω⃗p over ℛ as Ω:

[ Ω := Ω⃗p, ⃗p∈ℛ

′ and regard two paths ⃗p and ⃗p in Ω as equivalent if the Maslov index of the loop ′ ′ ′ ⃗p⃗p is even, where ⃗p is the reverse path of ⃗p. Assembling these two equivalence classes into a single space to obtain

P⃗p := Ω⃗p/∼, whose union as ⃗p ranges over ℛ forms a double cover

[ P⃗p =: P −→ ℛ. (3.2) ⃗p∈ℛ

Lemma 3.1. P → ℛ in (3.2) is trivial on each connected component of ℛ.

′ Proof. First observe that the relative Maslov indices of any two paths ⃗p and ⃗p connecting the same oriented Lagrangian subspaces can only diﬀer by an even number. So the parity of the relative Maslov index is independent of the reference path. On the other hand, for any choice of reference path 0, the parity of the relative Maslov index of a path ⃗p depends only on the (local) orientations of TxL and TyL, where ⃗p = (x, y). The latter property is clearly both open and closed since L is oriented. Hence the parity of the relative Maslov index is constant on components of ℛ.

Associated to each path ⃗p in Ω⃗p is the trivial real vector bundle

[ F⃗p := {t}×⃗p(t) −→ [0, 1]. t∈[0,1] CHAPTER 3. ORIENTABILITY OF THE MODULI SPACES 27

∼ n = Consider any trivialization ⃗p : [0, 1]×ℝ −→ F⃗p of the bundle F⃗p . The composition -1 (d) ∘ ⃗p ∣{0,1} induces framings on ℒx and ℒy, which give an embedding ⃗p ∗ from ∗ ∗ SO(n)×SO( V⃗p) to PSO(TL⊕ V )∣⃗p, making the following diagram commute:

' ∗ ⃗p ∗ (Spin(n)×Spin( V⃗p))/{±1} / PSpin(TL⊕ V )∣⃗p

∗ ∗ SO(n)×SO( V⃗p) / PSO(TL⊕ V )∣⃗p ⃗p ∗

There are two possible such '⃗p’s lifting ⃗p ∗, whose homotopy classes depend only on that of the trivialization ⃗p . It is not hard to see that the space of trivializations

⃗p of F⃗p which respect the orientation is homotopy equivalent to SO(n). So up to homotopy, there are two equivalence classes.

Denote the space of equivalence classes [⃗p ] of trivializations of F⃗p as T⃗p . Deﬁne

[ T := T⃗p −→ P.

⃗p

Finally, denote by Te⃗p the space of equivalence classes ['⃗p], and by Te the union of Te⃗p as ⃗p varies: [ Te := Te⃗p −→ P. (3.3)

⃗p We have 3 2 1 prX Te / T / P / ℛ / prX (ℛ) ⊆ X, in which Te → T is a double cover, and prX (ℛ) is the self-intersection of the immersed Lagrangian in the target manifold. Te can be considered as the space of ‘spin structures’ on the Lagrangian path space Ω. Now the family of operator

∂, −→ P deﬁned in (2.6) can be pulled back to Te. Let D → Te be this pull-back family. CHAPTER 3. ORIENTABILITY OF THE MODULI SPACES 28

Deﬁnition 3.2. Let D be a Fredholm operator between Banach spaces. The determinant line of D is deﬁned by

^max ^max ∗ det(D) := (ker D) ⊗ (coker D) .

Proposition 3.3 ([FOOO06, Prop. 51.1]). The determinant line det(D) of D de- scends to real line bundle Θ over prX (ℛ).

We pull back the line bundle Θ by prX to obtain the real line bundle

∗ lℛ := prX Θ −→ ℛ. (3.4)

This bundle plays an important role in the orientation of the moduli space. Deﬁne the line bundle L by

O ∗ L := det(D) ⊗ evlj lℛ, (3.5) j∈ℐ

where D := {D,p}p∈ℛ∣ℐ∣ is the family of Fredholm operators in (2.6). It is a line 1,p bundle over Bm+1, where

1,p 1,p n 2m+1 o Bm+1 := Wf (X, (L), ℐ)× ∂D ∖ Δ .

Let Oℳ be the orientation line bundle of the moduli space ℳm+1(J, ℐ, ). Notice that ∼ Oℳ = det(D) ⊗ Oℛ, where Oℛ is the orientation line bundle of ℛ.

3.2 Orientation of the moduli spaces

We are now ready to state the theorem.

Theorem 3.4. Let : L → X and ℛ be as perviously deﬁned in (1.1). Suppose that CHAPTER 3. ORIENTABILITY OF THE MODULI SPACES 29

1. L is spin or relatively spin, in particular, L is orientable,

2. ℛ is orientable,

3. the bundle Te → ℛ as deﬁned in (3.3) is trivial, then =∼ N ∗ Oℳ / j∈ℐ evlj lℛ ⊗ Oℛ . (3.6)

Further, ﬁxing a choice of the following data:

1. a spin or relatively spin structure on L, and

2. an orientation on ℛ,

3. a section s : ℛ → Te of the bundle Te → ℛ, determines a section of the bundle in (3.6).

Corollary 3.5. Under the assumptions of Theorem 3.4 above, the moduli space

ℳm+1(J, ℐ, ) is canonically orientable.

Note that a choice of a spin or relatively spin structure on L amounts to making the following choices:

1. an orientation on L,

2. an oriented vector bundle V on the 3-skeleton X[3] of X, and

∗ 3. a spin structure on (TL⊕ V )∣L[2] .

The spin structure then determines a homotopy class of trivialization of (TL⊕∗V ) over the the 1-skeleton L[1] of L that can be extended to the 2-skeleton L[2].

Proof of Theorem 3.4. For the first part of Proposition 3.4, so we adopt the proof of [FOOO06, Prop 51.6]) with a slight modification. The only difference is that we work with the space ℛ instead of its image in the target manifold X (as in [FOOO06]). CHAPTER 3. ORIENTABILITY OF THE MODULI SPACES 30

Remark 3.6. The case in [FOOO06] corresponds to the case where

prX : ℛ → prX (ℛ) ⊆ (L) is a trivial cover in our situation.

Recall that the orientation line bundle Oℳ of the moduli space ℳm+1(J, ℐ, ) is isomorphic to

det(D) ⊗ Oℛ.

We glue in the operators ∂ to D by a suitable partition of unity to get a Fredholm ,⃗pj operator D,. Then by a family version of the index sum formula, we have

∼ Y Ind D, = Ind ∂, ⊕ T⃗p ℛ × (Ind D) . (3.7) ⃗pj j T⃗p ℛ ( j )j j

It follows that the orientation of Ind D is determined by the orientations of Ind ∂ ⊕ ,⃗pj

T⃗pj ℛ and T⃗pj ℛ, where j ranges over ℐ. Each T⃗pj ℛ appears twice, so their eﬀects on orientation of Ind D cancel locally at ⃗pj. Since by assumption ℛ is orientable, any choice of an orientation gives a consistent way to orient the tangent spaces T⃗pℛ at every ⃗p ∈ ℛ. This implies the contributions of pairwise T⃗pj ℛ’s to the orientation in (3.7) cancel consistently for all j ∈ ℐ.

It remains to show that Ind D, has a canonical orientation. We need to show 1 1,p that any S -family Ind(D,)ut over any loop ut in Bm+1 can be given a consistent orientation. This is where we need to use the (relatively) spin structure on L and the assumption that Te → ℛ is trivial. So assume that Te → ℛ is trivial and we choose a section s : ℛ → Te. By the deﬁnition of Te, the section s determines the spin structure on the family of

Lagrangian subbundles (of the elliptic boundary value problem of D, in a consistent 1 way. Standard arguments then show that the S -family Ind(D,)ut has a consistent orientation induced by the one on Ind(D,)u0 . Chapter 4

Lagrangian Floer cohomology

4.1 A brief overview

In this section we investigate information which can be extracted from the moduli spaces ℳm+1(J, ℐ, ). This results in a theory of Lagrangian Floer cohomology. The corresponding theory for embedded Lagrangians have been studied extensively, for example, by Floer [Flo88a, Flo88b, Flo88c, Flo89], Robbin and Salamon [RS93, RS95], and Oh [Oh93a, Oh93b, Oh96]. In this setting, the (co)-chain complexes are generated by tansverse intersections between pairwise Lagrangians, graded by a suitable notion of Maslov index under favorable assumptions i. The (co)-boundary map is defined by counting holomorphic strips with boundaries lying on the relevant Lagrangians, with its ends at infinity converge to the intersections. This is done by considering the 0-dimensional components of the moduli spaces of holomorphic strips with the aforementioned boundary conditions. However, it often requires favorable assumptions in order for the (co)-boundary map to give the differential of a (co)-homology theory. When the Lagrangians do not interect transversely (this includes the case when there is only one embedded Lagrangian), there are various approaches in the lit- erature: Cornea-Lalonde [CL06], Po´zniak[Po´z99],Seidel [Sei08] and others. We will

i For example the c1 of the manifold being trivial, or the Maslov class of the Lagrangians being trivial, so that the Maslov index is independent of the homotopy class of the holomorphic strip. One may also work with an enlarged ring (Novikov ring) in which an extra variable is introduced to keep track of Maslov indices.

31 CHAPTER 4. LAGRANGIAN FLOER COHOMOLOGY 32

particularly focus on the work of Fukaya et. al. [FOOO06] in the Morse-Bott situation. In this setting, the (co)-chain complexes are (co)-chains in the clean intersections of the Lagrangians. The (co)-boundary map is defined by considering moduli spaces of holomorphic disks with boundary on the Lagrangians, in which the moduli spaces are cut down to the right dimension by intersecting them with (co)-chains in the intersections. Again there are various technical difficulties that one needs to deal with in order to get a well-defined. These issues will be discussed later in the context of some specific examples. Finally, Floer cohomology of Lagrangian immersions with transverse double point self intersections was considered in [AJ08]. One expects the theory should extend to Lagrangian immersions with clean intersections. Remark 4.1. Lagrangian Floer theory is not always defined as various obstructions may arise. There are various technical issues one must resolve in order for the theory to be well-defined, namely

1. compactness and transversality of the moduli space,

2. orientability of the moduli space,

3. whether (co)-boundary map ∂ deﬁnes a (co)-chain complex.

The compactness issue can be tackled by Gromov’s compactness. Away from multiply covered disks, standard analysis (cf. [MS04]) shows that transversality can be achieved by using a generic J. Furthermore, if, for example, L is monotone ii, then dimension counts show that multiply-covered disks do not cause problems. In general, there are several ways to deal with the issue of transversality. One is by perturbing the operator by an inhomogeneous term to achieve transversality. The other is by considering domain-dependent J’s. One can also use multivalued perturbations, or work with structures like polyfolds or Kuranishi structures. For the (co)-boundary maps, bubbling is a serious technical issue. For instance, disks of Maslov index 0 may cause problems to ∂2 = 0. Suitable assumptions are required to rule out such boundary bubblings in order to obtain (co)-chain complex.

ii This means L = ! for some > 0. CHAPTER 4. LAGRANGIAN FLOER COHOMOLOGY 33

For example in the embedded case, one may assume the Lagrangians are monotone with a minimal Maslov index of 2. The monotone condition is used to rule out multiply-covered disk, while the minimal Maslov index ensures that boundary disk bubbling occurs in codimension 2 and so does not contribute to the boundary of the moduli space. This also applies to Lagrangian immersions that are also covering spaces, as long as the image is embedded, monotone with minimal Maslov index at least 2. In the later section we will study some examples and compute their Floer cohomologies in this situation. Chapter 5

Examples and computations

In the following sections we will study two examples involving ℝPn. The ﬁrst one is sphere double covers of ℝPn and the second one is its orientation covers.

5.1 Sphere double covers of ℝPn in ℂPn

Consider the sphere double cover : Sn → ℝPn ,→ ℂPn given as the quotient space of the antipodal map on the sphere Sn. In particular, the map is a Lagrangian immersion with respect to the standard K¨ahlerstructure on ℂPn. With our earlier n n notations, S ×S is the disjoint union, ΔSn ∐ ΔSn , of the diagonal and the anti- n n n diagonal of S ×S respectively. Both ΔSn and ΔSn are diﬀeomorphic to S in this case. The aim of this example is to compute the Lagrangian Floer cohomology of the sphere double cover : Sn → ℝPn. First we survey here some well-known facts about ℝPn which will be used later. These can be found, for example, in Oh’s papers [Oh93a, Oh93b]. First of all, ℝPn is monotone, meaning that

I = I!, for some > 0,

34 CHAPTER 5. EXAMPLES AND COMPUTATIONS 35

where I and I! are homomorphisms

n n I,I! : 2(ℂP , ℝP ) −→ ℤ, ℝ

respectively, given by the Maslov index and the symplectic area. Denote by ΣℝPn the positive generator of the abelian subgroup

n n [∣2(ℂP ,ℝP )] ⊆ ℤ.

Then

ΣℝPn = n + 1 is the minimum Maslov number for ℝPn. n n n Since : S → ℝP is a covering space, 2(ℝP ) is trivial, and this leads to the short exact sequence of homotopy groups

n n n ∂∗ n 0 / 2(ℂP ) / 2(ℂP , ℝP ) / 1(ℝP ) / 0

n n and hence 2(ℂP , ℝP ) splits as

n n ∼ 2(ℂP ) ⊕ 1(ℝP ) = ℤ ⊕ ℤ/2ℤ.

n n In particular, there are two classes 1 and 2 in 2(ℂP , ℝP ) whose images under the Maslov index homomorphism I is the generator ΣℝPn, and whose images under n ∂∗ are non-trivial in 1(ℝP ). These two classes j represent the two ‘halves’ of the complex line ℂP1 which are exchanged by the anti-symplectic involution

n n : ℂP −→ ℂP induced by complex conjugations on its charts, whose ﬁxed points set Fix() is exactly the Lagrangian submanifold ℝPn. CHAPTER 5. EXAMPLES AND COMPUTATIONS 36

zk 5.2 The moduli spaces ℳ2 ( ) with one jump

Here we discuss the moduli spaces involved in the construction of the Lagrangian Floer cohomology discussed later in section 5.3. Although a priori, one can consider moduli n n spaces of pseudo-holomorphic disks representing any relative classes in 2(ℂP , ℝP ), for degree reasons which will be made clear in Section 5.3, only the classes 1 and

2 corresponding to the minimal Maslov index are relevant in the definition of the differentials of the Lagrangian Floer cohomology. 2 2 n n Suppose u :(D , ∂D ) → (ℂP , ℝP ) is a map which represents the class j. Since 2 n n u∗[∂D ] is non–trivial in 1(ℝP ), u∣∂D2 cannot be lifted to S as a continuous map on ∂D2. On the other hand, (discontinuous) lifts in the sense of Definition 2.4 can be defined once jump points are introduced to divide the boundary of D2 into arcs. In fact, it requires at least one jump point on ∂D2 for lifts in Definition 2.4 to exist for any map u representing the classes 1 and 2. There is an evaluation map evaluating at this marked point to the n-dimensional manifold ℛ, which is diffeomorphic to the n n anti-diagonal ΔSn of S ×S . As we will see later, for dimension reasons, at least one more marked point is required to define the desired Floer cohomology. With that understood, we introduce another marked point on the boundary of the disk. However topologically, lifts with two jumps do not exist. It follows that the moduli spaces with two jump points are empty. Hence we only need to consider moduli spaces ℳ2(J0, {k} , ) with one jump, where k = 0 or 1 and is a class corresponding to the minimal Maslov index. Here J0 denotes the the standard complex structure on ℂPn. For simplicity, in the following

zk we will denote by ℳ2 ( ) the moduli spaces ℳ2(J0, {k} , ). Remark 5.1. There is a forgetful map which forgets lifts of holomorphic disks

F zk zk ℳ2 ( ) / M 2 ( ),

zk where M 2 ( ) is the moduli space of pseudo–holomorphic disks with boundary on n ℝP without lifts, but has zk as a special marked point. For a non-trivial class , the CHAPTER 5. EXAMPLES AND COMPUTATIONS 37

forgetful map F gives rise to a non-trivial double cover

F zk zk ℤ/2ℤ / ℳ2 ( ) / M 2 ( ).

5.2.1 Regularity of D∂J0

In this section we discuss briefly the regularity of the operator D∂J0 and demonstrate why coker Du∂J0 is trivial for any holomorphic disk in the class j with minimum Maslov index n+1. The analysis here is a slight modification of those in [Oh93b]. This justifies the use of the standard complex structure J0 to calculate Floer cohomology in the later section.

Let us first explicitly compute the linearization Du∂J0 at a disk u representing the class j. For simplicity, denote by Du the linearized operator Du∂J0 . Here, ℂPn is endowed with the Fubini-Study Kählerform ! with the compatible standard n complex structure J0. With respect to this Kählerstructure, ℝP is a totally real submanifold. We denote by ∇ the Levi-Civita connection of the Fubini-Study metric

⟨⋅, ⋅⟩ = !(⋅,J0⋅). This connection is torsion-free and Hermitian with respect to the complex structure J0. In holomorphic coordinates z = s + it on D2,

∂J0 u = ∂su + J0(u)∂tu = 0 Z n 2 2 u(z) ⊆ ℝP for z ∈ ∂D , and ∣∇u∣ < ∞

We know by elliptic regularity that u is C∞. For any ∈ W k,2, where W k,2 is some n 2 appropriate Sobolev space, with (z) ⊆ Tu(z)ℝP for z ∈ ∂D , we have

Du() = ∇ ∣=0∂J0 (u ) CHAPTER 5. EXAMPLES AND COMPUTATIONS 38

d where u0 = u and d ∣=0u = .

∇ ∣=0∂J0 (u ) = ∇ ∣=0 (∂su + J(u )∂tu )

= ∇ ∂su ∣=0 + ∇ J0(u )∣=0∂tu + J0(u)∇ ∂tu ∣=0

= ∇s∂ u ∣=0 + J0(u)∇t∂ u ∣=0

= ∇s + J0(u)∇t

which implies that Du = ∇s + J0(u)∇t. The operator Du is Fredholm and has a closed range. The goal is to show that its range is dense and thus coker Du is trivial. ∗ ∗ This is equivalent to the triviality of ker Du. The adjoint Du is characterized by the equation ∗ ⟨, Du⟩2 = ⟨Du, ⟩2 for any , ∈ W k,2 with respect to the L2-norm. Then we have Z ⟨Du, ⟩2 = ⟨∇s + J0(u)∇t, ⟩ D2 Z = ⟨∇s, ⟩ + ⟨J0(u)∇t, ⟩ D2 Z Z = − ⟨, ∇s⟩ + ⟨(z), (z)⟩ Re z dl D2 ∂D2 Z Z − !(, ∇t) + !((z), (z)) Im z dl D2 ∂D2 Z = ⟨, −∇s + J0(u)∇t⟩ D2 Z + ⟨(z), (Re z)(z) − (Im z)J0(u)(z)⟩ dl ∂D2 Z Z = ⟨, −∇s + J0(u)∇t⟩ + ⟨(z), z(z)⟩ dl. D2 ∂D2

with an appropriate trivialization of bundle. The third equality is obtained by inte- gration by part, and the dl in the last three equalities is the length measure on the CHAPTER 5. EXAMPLES AND COMPUTATIONS 39

2 2 ∗ boundary ∂D . The L inner product ⟨⋅, ⋅⟩2 allows us to identify coker Du with ker Du, and we may assume that for any ∈ coker Du, ⟨Du, ⟩2 = 0 for all . Therefore we obtain the following adjoint problem

2 (−∇s + J0(u)∇t) = 0 for ∈ L (5.1) n⊥ 2 z(z) ∈ Tu(z)ℝP for all z ∈ ∂D .

Again any ∈ coker Du is smooth by elliptic regularity. For any given ∈ coker Du, we reﬂect the holomorphic disk u across the boundary of the domain D2 and apply removal of isolated singularities to obtain a holomorphic map

1 n w : ℂP → ℂP .

Similarly, we can reﬂect across the boundary to get a smooth vector ﬁeld

2 n X : S −→ T ℂP

n n such that ∥X∥1,2 is ﬁnite and ∘ X(z) = w(z), where : T ℂP → ℂP . Next we complexify the tangent bundle T ℂPn, and denote its complexiﬁcation by n n TℂℂP = T ℂP ⊗ ℂ. It decomposes to

n TℂℂP = E ⊕ E a direct summand between a holomorphic vector bundle E and its conjugate bundle E with the opposite complex structure to E. The bundle E is isomorphic to the Hermitian bundle T ℂPn. Its Hermitian structure allows us to identify the conjugate ∗ ∼ ∗ n bundle E with the dual E = T ℂP as a holomorphic vector bundle.

Since w is J0-holomorphic, the pull-back by w respects the splitting

∗ n ∗ ∗ w TℂℂP = w E ⊕ w E, where the pull-back bundle w∗E is isomorphic to w∗T ℂPn and w∗E is isomorphic to CHAPTER 5. EXAMPLES AND COMPUTATIONS 40

w∗T ∗ℂPn. Equation (5.1) then becomes

−∇s + i∇t = 0 which says that deﬁnes an anti-holomorphic section of w∗E, or in other words, a ∗ ∼ ∗ ∗ n holomorphic section of w E = w T ℂP . Recall the following lemma from complex geometry whose proof can be found, for example, in [Oh93b, Lemma 2.6].

Lemma 5.2. Let : ℂP1 → ℂPn be a non-constant holomorphic map with respect to the usual complex structures on ℂP1 and ℂPn. Then the Grothendick splitting

∗ n E := w T ℂP = L1 ⊕ ⋅ ⋅ ⋅ ⊕ Ln,

where the Li’s are holomorphic line bundles, has the property that c1(Li) > 0 for all i = 1, . . . , n. In particular, there is no non-trivial holomorphic section on E∗ ∼= w∗ (T ∗ℂPn).

However, if is non-trivial, then this contradicts Lemma 5.2 above. Therefore, coker Du must be trivial.

zk 5.2.2 Orienting the moduli spaces ℳ2 ( j )

zk As we will see later in Section 5.3, the moduli spaces ℳ2 ( j ) are involved in the construction of Lagrangian Floer cohomology. In general, the cochain complexes of Lagrangian Floer cohomology are defined only with ℚ coefficients, due to the presence of multiply-covered disks. So one cannot pass to ℤ/2ℤ coefficients and have to deal with the problem of orientation. However, as we will discuss later, the differentials of Lagrangian Floer cohomology for ℝPn only involve the classes of disks with minimal Maslov index. Such disks are never multiply-covered, and this allows us to use ℤ coefficients to define the cochains of Floer cohomology. Hence for later applications, we study the problem of orienting the moduli spaces CHAPTER 5. EXAMPLES AND COMPUTATIONS 41

zk ℳ2 ( j ) by applying the ideas discussed in §3. Given a map

2 2 n n u :(D , ∂D ) → (ℂP , ℝP ), with an associated lift ue and a jump at zk, we are going to describe explicitly a way zk to orient the tangent space of the moduli space ℳ2 ( j ) at this point.

At each jump point zk whose image under u is pk, we choose a path pk in ori the space of oriented Lagrangian Grassmanian Lag (Tpk X), which can match nicely with the boundary of the map u at zk to give a loop in the pull-back of the La- grangian Grassmanian of TX to (D2, ∂D2). We then obtain a well-deﬁned bundle pair (E,F ) → (D2, ∂D2), where E is a complex vector bundle and F is a totally real subbundle. Associated to the path is the Cauchy-Riemann operator ∂ , whose Fredhlom pk ,pk index is denoted by k. The value of k depends on the number of windings of the path pk between the two corresponding Lagrangian subspaces at the point u(zk). n n More precisely, if u represents the class in 2(ℂP , ℝP ), then

2 k = w1(u∣∂D2∗[∂D ]) ≡ w1(∂∗ ) (mod 2) (5.2)

= I( ) + k,

where is the Maslov index of the loop . Specializing to the class j,

= n + 1 + k.

In the generic situation where is represented by a pseudo-holomorphic disk, possibly after a perturbation, since the Lagrangian is monotone, must be a positive multiple of the minimal Maslov index n+1. Together, it follows that in this situation

k is either 0 or a positive multiple of n + 1. On the other hand, from (5.2), it is not hard to deduce that

k ≡ n + 1 (mod 2),

since the number of windings of Lagrangian subspaces in pk should be even when CHAPTER 5. EXAMPLES AND COMPUTATIONS 42

ℝPn is orientable, and odd in the other case.

zk To deﬁne an orientation on ℳ2 ( j ), we need to trivialize the totally real sub- 2 bundle F → ∂D . A necessary condition for the later is an even Maslov index , which is always true since

= n + 1 + k ≡ 0 (mod 2).

Then we can apply the method in Chapter 3 to orient the moduli spaces.

5.3 Morse-Bott Floer cohomology of the sphere covers of ℝPn

In this section we investigate an extension of Morse-Bott type Lagrangian Floer cohomology in [FOOO06] to our case. Since locally, a cleanly intersecting immersed Lagrangian just resembles the Morse-Bott situation, we should expect a similar theory for this type of Lagrangian immersions. Intuitively, the information of self-intersections of the Lagrangian immersion is contained in the ﬁber product L×L. Therefore, to construct the desired Lagrangian Floer cohomology in this case, we would need a cochain complex which represents the cohomology of L×L.

We will work with smooth simplicial (co)chains on L×L as in [FOOO06]. Denote by Δn−k the co-dimension k simplex deﬁned as

n−k+1 (x0, . . . , xn−k) ∈ ℝ ∣ xi ⩾ 0, x0 + ⋅ ⋅ ⋅ + xn−k = 1 .

For any smooth map f :Δn−k → L×L, deﬁne its cohomology degree as k. k k Let C := C (L×L; ℤ) be the free abelian group generated by such maps. To- gether with the usual coboundary maps, they form the cochain complex that we desire. A general element in Ck will be denoted by the pair [P, f], where P is a codimension k simplicial complex, and f : P → L×L is a (piece-wise) smooth map. This will often be abbreviated as P unless there is a risk of confusion. CHAPTER 5. EXAMPLES AND COMPUTATIONS 43

Remark 5.3. Technically, since intersection theory will be used implicitly, in general one should focus on (co)chains which intersect our moduli spaces transversely. How- ever, in the case of ℝPn, our moduli spaces satisfy the desired transversality properties. So by perturbations if necessary, we can make the (co)chains intersect transversely with our moduli space.

5.3.1 Novikov Ring

In deﬁning Lagrangian Floer cohomology, one may introduce a ring of formal power series known as the Novikov ring to keep track of energy and Maslov indices of disks. More precisely,

Deﬁnition 5.4. Let T and e be formal parameters. The Novikov ring is deﬁned as

( ∞ ) R X i ki Λ = aiT e ∣ ai ∈ R, ki ∈ , i ∈ , lim i = ∞ nov ℤ ℝ i→∞ i=0 where R is any commutative ring.

In deﬁnition 5.4, the terms ek and T formally keep track of holomorphic disks R with Maslov index 2k and symplectic area . The commutative ring Λnov is graded by setting deg(aT ek) = 2k.

R There is the following graded subring of Λnov.

Deﬁnition 5.5. Let T and e be formal parameters. We set

( ∞ ) R X i ki Λ = aiT e ∣ ai ∈ R, ki ∈ , i 0, lim i = ∞ 0,nov ℤ ⩾ i→∞ i=0 where R is any commutative ring.

For monotone Lagrangian submanifolds such as ℝPn, the Maslov index of any disk is proportional to its energy and is positive. In this case, a reduced Novikov ring can used in which the parameter T keeping track of energy is omitted. In the CHAPTER 5. EXAMPLES AND COMPUTATIONS 44

following discussions, this reduced Novikov ring Will be used entirely. We will abuse the previous notations by setting

Deﬁnition 5.6.

( ∞ ) R X ki Λ = aie ∣ ai ∈ R, ki ∈ 0, lim ki = ∞ 0,nov ℤ⩾ i→∞ i=0

R R There is a filtration on Λnov and thus on the subring Λ0,nov by the energy of disk. Specializing to the case of monotone Lagrangian submanifolds, we define this filtration by the family of subsets

( ∞ ) k R X ki F Λ0,nov = aie ∣ 2ki ⩾ k . i=0

5.3.2 Floer coboundary operators

In this section we are going to discuss the deﬁnition of the coboundary operator of Morse-Bott Floer cohomology.

Deﬁnition 5.7. For any co-dimension ∗ chain [P, f] in C∗, deﬁne

(P ) := evl0*(ℳ2( )×evl1,f P ) for ∕= 0, n 0(P ) := (−1) ∂P, where ∂ is the usual co-boundary operator

∗ n n ∂ ∗+1 n n C (S ×S ; ℤ) / C (S ×S ; ℤ),

zk and ℳ2( ) is the disjoint union ∐kℳ2 ( ). Again evl0*(ℳ2( )×evl1,f P ) will be abbreviated as evl0*(ℳ2( )×evl1 P ).

Remark 5.8. For any co-chain P in Cl and non-trivial , the ﬁber product

evl0*(ℳ2( )×evl1 P ) CHAPTER 5. EXAMPLES AND COMPUTATIONS 45

has a Kuranishi structure. Its virtual fundamental chain can be regarded as a co-chain

l−I( )+1 l−I( )+1 in C : given a (l − I( ) + 1)-dimensional chain Q in C , Z ∗ ∗ ⟨Q, (P )⟩ := evl0 (PD(Q)) evl1 (PD(P )), (5.3) vir ⌣ [ℳ2( )]

n where the virtual dimension of ℳ2( ) is n + I( ) − 1. On the other hand, since ℝP is monotone, I( ) is always a positive multiple of ΣℝPn = n + 1. For degree reason,

n + 1 ⩽ I( ) ⩽ l + 1.

Hence, in order for the pairing (5.3) to be non-zero,

l = n and I( ) = n + 1.

n n There are exactly two classes 1 and 2 in 2(ℂP , ℝP ) with Maslov indices n + 1.

i ℤ By extending linearly over the Novikov ring Λ0,nov , we obtain a co-boundary operator on the complex

∗ n n ℤ ∼ ∗ n n ℤ C (S ×S ;Λ0,nov) = C (S ×S ; ℤ) ⊗ℤ Λ0,nov.

k ∗ n n ℤ Deﬁnition 5.9. For any co-chain P ⊗ e in C (S ×S ;Λ0,nov), we set

X I( )/2 (P ) = (P ) ⊗ℤ e . n n ∈2(ℂP ,ℝP )

The degree of P ⊗ ek is deﬁned as

deg(P ⊗ ek) = deg P + 2k. (5.4)

l ℤ With the grading deﬁned in (5.4) above, given any co-chain P in C ⊗ Λ0,nov with

i ℚ In general, the Novikov ring Λ0,nov is used if multi-valued perturbations of the Kuranishi structure are needed instead of single-valued ones to achieve the relevant transversality of the perturbed moduli spaces. CHAPTER 5. EXAMPLES AND COMPUTATIONS 46

l−I( )+1 ℤ degree l, its image (P ) under the operator is a co-chain in C ⊗Λ0,nov with a degree given by

deg (P ) = l − I( ) + 1 + I( ) = l + 1.

∗ ℤ So the degree of the map is 1. deﬁnes a co-boundary operator on C ⊗ Λ0,nov. Remark 5.10. Compared with [FOOO06] and [AJ08], the gradings of the complexes found there include a term related to the added path at each jump point. In the Morse-Bott case of [FOOO06], this term is the relative Maslov index of the added path. While in the case of [AJ08] of Lagrangian immersions with transverse self- intersections, it is the Fredhlom index of the Cauchy-Riemann operator associated to the path. In both situations, these added paths are required in order to have a well- deﬁned loops in the Lagrangian Grassmanian for any holomorphic disk with boundary on the respective Lagrangians. On the other hand, for the example consided here, the image of the Lagrangian immersion is an embedded Lagrangian submanifold, and

I( ) is already well-deﬁned depending only on the class , even without the paths at the jump points. In light of Remark 5.8, the operator simpliﬁes to

Σ n/2 ℝP = 0 + ( 1 + 2 ) ⊗ e . (5.5)

l ℤ One can show that ∘ = 0 directly as follows. For any co-chain P in C ⊗ Λ0,nov,

Σ n/2 Σ n/2 ℝP ℝP ∘ (P ) = 0 ( 1 + 2 ) P ⊗ e + ( 1 + 2 ) 0P ⊗ e (5.6) 2 X Σ n ℝP + i j P ⊗ e . i,j=1

First observe that terms of the form i j P on the right hand side of (5.6) are always n ℤ zero, since the operators j is only non-trivial on C ⊗ Λ0,nov, in which co-chains are 0 ℤ mapped to C ⊗ Λ0,nov.

Next consider terms of the form 0 j (P ). Notice that j (P ) can only be non- n ℤ 0 ℤ zero if P is a co-chain in C ⊗ Λ0,nov, whereas it is a co-chain in C ⊗ Λ0,nov. But CHAPTER 5. EXAMPLES AND COMPUTATIONS 47

n n+1 ∂ : C −→ C is trivial. Hence the terms 0 j (P ) are always zero.

Finally for the remaining terms j 0(P ), we only have to consider the case when n−1 ℤ n n P ∈ ℂ ⊗ Λ0,nov. So P is a co-dimension n − 1 chain, or a path, in ΔS ∐ΔS . Let

: [0, 1] → ΔSn ∐ΔSn be the path it represents. Then it follows that

j (∂P ) = evl0* (ℳ2( j )×evl1, (∂P )) n = (−1) evl0* (∂ℳ2( j )×evl1, P ) = 0.

Therefore we obtain the following lemma.

Lemma 5.11. For all i, j = 1, 2,

0 ∘ j = j ∘ 0 = 0

i ∘ j = 0.

In particular, this implies that ∘ = 0 and Σ n ∘ Σ n = 0. ℝP ℝP

5.3.3 Spectral sequence and Floer cohomology

The positive generator ΣℝPn of I gives rise to the following ﬁltration of the coboundary operator,

∞ X kΣ n/2 Σ n/2 ℝP ℝP = kΣ n ⊗ e = 0 + Σ n ⊗ e , (5.7) ℝP ℝP k=0 where k is the formal sum of ’s with Maslov index k. In the subsequent discussions, ∗ ℤ ∗ ℤ the complexes C ⊗ Λ0,nov will be denoted as CF , with coeﬃcients ring Λ0,nov unless otherwise stated. In particular, CF ∗ is doubly graded by the gradings on C∗ and the ℤ Maslov index on Λ0,nov. We set

p,q p q/2 CF := C ⊗ℤ e . CHAPTER 5. EXAMPLES AND COMPUTATIONS 48

Then (CF ∗,∗, ) deﬁnes a spectral sequence for Lagrangian Floer cohomology

∗,q p+q n n ℤ Hp (CF , ) ⇒ HF (S ×S ;Λ0,nov).

Proposition 5.12. Suppose n ≡ 1 or 2 (mod 4). Then there is a unique non-zero diﬀerential n n n 0 n n n+1 : H (S ×S ; ℤ) −→ H (S ×S ; ℤ) which is multiplication by ±2. In these cases,

∗ n n 0 n n Σ n ℤ ∼ ℤ ℝP ℤ HF (S ×S ;Λ0,nov) = H (S ×S ; ℤ) ⊗ Λ0,nov/2F Λ0,nov

Σ n ⊕2 ∼ ℤ ℝP ℤ = Λ0,nov/2F Λ0,nov .

On the other hand, if n ≡ 0 or 3 (mod 4), the diﬀerential = 0. Hence we have,

∗ n n ℤ ∼ ∗ n n ℤ HF (S ×S ;Λ0,nov) = H (S ×S ; ℤ) ⊗ Λ0,nov ∼ ℤ ⊕2 ℤ ⊕2 = Λ0,nov ⊕ Λ0,nov .

Remark 5.13. The diﬀerential n+1 in Proposition 5.12 is not always non-trivial in odd dimensions, which is the case in [FOOO06, Theorem 44.24]. This is mainly due to the deﬁciency of distinct spin or relatively spin structures on Sn.

Proof of Proposition 5.12. As we discussed previously, for degree reasons, the only non-trivial contributions to the diﬀerential come from 0 and

n n 0 0 n n n n j : CF (ΔS ) ⊕ CF (ΔS ) −→ CF (ΔS ) ⊕ CF (ΔS ), j = 1, 2,

where disks representing the classes j have the minimal Maslov index ΣℝPn, so

= 0 + 1 + 2 .

By deﬁnition, the evaluation maps

zk n n evlj : ℳ2 ( ) −→ ΔS ∐ΔS CHAPTER 5. EXAMPLES AND COMPUTATIONS 49

evaluate to ΔSn ≈ ℛ only if j = k.

First consider the case where z1 is the jump point, evl1 evaluates z1 to the antidiagonal ΔSn , and the evaluation map evl0 evaluates z0 to the diagonal ΔSn . Given any n n co-dimension n chain P in CF (ΔS ), it then follows that j (P ) is a co-dimension 0 0 n chain in CF (ΔS ). So in this case, the diﬀerential j deﬁnes a map

n 0 n n j : CF (ΔS ) −→ CF (ΔS ),

n n in which the restriction of j to CF (ΔS ) is trivial. It follows that for each j, the image of P under j is

z1 (n+1)/2 j (P ) = {evl0∗ (ℳ2 ( j )×evl1 P )} e , (5.8)

(P ) is then the sum over j of the expressions in (5.8).

Lemma 5.14. Consider the product evl = evl0×evl1 of the two evaluation maps:

z1 n n evl : ℳ2 ( j ) −→ ΔS ×ΔS for j = 1, 2.

z 1 n n n n Then evl* ([ℳ2 ( j )]) is the fundamental cycle ΔS ×ΔS of ΔS ×ΔS .

Similarly, when z2 is the jump point,

z2 n n evl : ℳ2 ( j ) −→ ΔS ×ΔS for j = 1, 2,

z 2 n n n n and evl* ([ℳ2 ( j )]) gives the fundamental cycle ΔS ×ΔS of ΔS ×ΔS .

Proof of Lemma 5.14. Given any two distinct points p and q in ℝPn which is embedded in ℂPn in the standard way, there is a non-constant holomorphic map v : ℂP1 → ℂPn of degree one mapping 0 to p and ∞ to q. Here we identify the upper hemisphere of ℂP1 with the upper half space in ℂ. This map is unique up to the action of the group ℂ∗ = ℂ ∖ {0} of automorphisms of ℂP1 ﬁxing the points 0 and ∞. In fact, such automorphism acts on v by ⋅ v(z) = v(z), and it ﬁxes v if and only if = 1. The degree one map v is not multiply covered, and by re-parametrization if necessary, we may assume that v maps ℝ ∪ {∞} to ℝPn. Then the restrictions of v to CHAPTER 5. EXAMPLES AND COMPUTATIONS 50

z1 z1 the upper and the lower half spaces deﬁne elements in ℳ2 ( 1 ) and ℳ2 ( 2 ) re- n spectively, with well-deﬁnes lifts which map z1 to the pair of antipodal points on S corresponding to q. So there exits v ∈ ℳ2( j ) such that evl(v) = (p, q).

z1 Conversely, any element v ∈ ℳ2 ( j ) defines a degree one rational curve by the reflection principle. This proves the first part of Lemma 5.14. The second part of Lemma 5.14 follows from a similar argument as before.

As a consequent of Lemma 5.14, the expression in (5.8) becomes

(n+1)/2 n j (P ) = ± [ΔS ] e , (5.9)

z1 where the sign depends on the orientation of the moduli space ℳ2 ( j ). We will see shortly how the sum 1 + 2 of the diﬀerentials depends on the orientations of the

z1 moduli spaces ℳ2 ( j ). Consider the anti-symplectic involution

n n : ℂP −→ ℂP

n whose ﬁxed points set Fix() is the Lagrangian submanifold ℝP , such that ∗( 1) = z1 2. Let u1 be elements in ℳ2 ( 1 ) representing 1. By identifying the unit disk 2 2 (D , ∂D ) with the upper half space in ℂ and composing the map u1 with , we obtain the map

u2(z) = ( ∘ u1)(z) (5.10)

z1 z1 which deﬁnes an element in ℳ2 ( 2 ). The orientation of the moduli space ℳ2 ( j ) is then determined by an orientation of the elliptic complex

2 2 2 0,1 Duj ∂ :Γ D , ∂D ; Ej,Fj −→ Γ D ;Λ ⊗ Ej

where Fj is a totally real subbundle of Ej which is the pull-back by uj of the tangent bundle of ℂPn, and the sections Γ are taken over appropriate Sobolev norms. CHAPTER 5. EXAMPLES AND COMPUTATIONS 51

The diﬀerential of deﬁnes a bundle map

d (E1,F1) −−−→ (E2,F2) ⏐ ⏐ ⏐ ⏐ y y (D2, ∂D2) −−−→ (D2, ∂D2) c covering the involution induced by the complex conjugation in ℂ in the upper half space model. On the other hand, a trivialization

n Φ:(E1,F1) −→ (ℂ , Λ)

2 2 of the bundle pair (E1,F1) → (D , ∂D ) naturally induces a trivialization

n Φ:(E2,F2) −→ (ℂ , Λ), where Λ and Λ are loops of Lagrangian subspaces of ℂn induced by by the trivializations. These two loops are related by

2 Λ(z) = Λ(c(z)) for z ∈ ∂D .

Together we obtain a bundle map

Φ∘d∘Φ−1 (ℂn, Λ) −−−−−−→ (ℂn, Λ) ⏐ ⏐ ⏐ ⏐ y y (D2, ∂D2) −−−→ (D2, ∂D2) c covering the complex conjugation, such that for any z ∈ ∂D2,

−1 n n Φ ∘ d ∘ Φ (z):(ℂ , Λ(z)) −→ (ℂ , Λ(c(z)) = Λ(z)) is the complex conjugation in ℂn with ﬁxed points set the Lagrangian subspace Λ(z). Recall that to orient the moduli space, we ﬁrst deform the metric on D2 by CHAPTER 5. EXAMPLES AND COMPUTATIONS 52

shrinking a concentric circle near the boundary so that the disk D2 degenerates to the union of D2 and ℂP1 glued at a point. We also let the trivialization Φ deforms so n 2 that Λ(z) becomes ℝ for any z on ∂D . In the process, the bundle pair (Ej,Fj) → (D2, ∂D2) deforms to one, such that it deﬁnes a holomorphic vector bundle E over ℂP1 and a bundle pair (Cn, ℝn) over the disk (D2, ∂D2), for j = 1, 2. After deforming the operators Duj ∂, the problem of orientation reduces to orienting the space

2 2 n n 1 ΓH (D , ∂D ; ℂ , ℝ )×ΓH (ℂP ; E),

where ΓH denotes holomorphic sections of the corresponding bundle pairs. The involution then induces a map

2 2 n n 1 :ΓH (D , ∂D ; ℂ , ℝ )×ΓH (ℂP ; E) 2 2 n n 1 −→ ΓH (D , ∂D ; ℂ , ℝ )×ΓH (ℂP ; E), which restricts to the usual complex conjugation on ℂn. Notice that

2 2 n n ∼ n 1 ∼ /2 ΓH (D , ∂D ; ℂ , ℝ ) = ℝ and ΓH (ℂP ; E) = ℂ , where ⎧ ⎨n + 2, if n is even; = ( j) + 1 = ⎩n + 1, if n is odd.

1 It is not hard to see that is orientation preserving on ΓH (ℂP ; E) if and only if

/2 is even. By deﬁnition,

n o z1 z1 22 ℳ2 ( j ) = ℳf ( j )× ∂D ∖ Δ / ∼, where Δ is the subset of (∂D2)2 in which two marked points coincide. Since complex conjugation in ℂ reverses the orientation of the boundary ∂D2 of the unit disk, it follows that is orientation preserving if and only if

1 1 2 = 2 (n + 1 + 1) + 2 ≡ 0 (mod 2). CHAPTER 5. EXAMPLES AND COMPUTATIONS 53

Equivalently, this happens when

n ≡ 2 (mod 4) when n is even, n ≡ 1 (mod 4) when n is odd.

Therefore we conclude by (5.9) that,

⎧ (n+1)/2 ⎨±2 [ΔSn ] e when n ≡ 1 or 2 (mod 4), ( 1 + 2 )(P ) = (5.11) ⎩0 when n ≡ 0 or 3 (mod 4).

Similarly, When the jump point is z0, evl1 evaluates z1 to the diagonal ΔSn , and

n evl0 evaluates z0 to the antidiagonal ΔS . In this case, the diﬀerential j is the map

n 0 n n j : CF (ΔS ) −→ CF (ΔS ).

n n For any given co-dimension n chain P in CF (ΔS ), its image under j is then

z0 (n+1)/2 j (P ) = {evl0∗ (ℳ2 ( j )×evl1 P )} e . (5.12)

Again by (5.9), we obtain

⎧ (n+1)/2 ⎨±2 ΔSn e when n ≡ 1 or 2 (mod 4), ( 1 + 2 )(P ) = (5.13) ⎩0 when n ≡ 0 or 3 (mod 4).

Therefore, (5.11) and (5.13) together imply that the diﬀerential

n n n 0 n n n+1 : H (S ×S ; ℤ) −→ H (S ×S ; ℤ) is non-trivial and is the multiplication by ±2 as stated in Proposition 5.12. CHAPTER 5. EXAMPLES AND COMPUTATIONS 54

Consider the E0 page of the spectral sequence.

CF 0,2n+2 / ⋅ ⋅ ⋅

CF 0,n+1 / ⋅ ⋅ ⋅ / CF n,n+1

0 CF 0,0 / CF 1,0 / ⋅ ⋅ ⋅ / CF n,0

To obtain the E1 page, we take homology of the horizontal arrows. Let

p,q p n n q/2 H := H (S ×S ; ℤ) ⊗ e .

We have,

H0,2n+2 0 ⋅ ⋅ ⋅ O n+1 H0,n+1 0 ⋅ ⋅ ⋅ 0 Hn,n+1 O n+1 H0,0 0 ⋅ ⋅ ⋅ 0 Hn,0

From this, it follows that the spectral sequence degenerates on the E2 page. Therefore, when n ≡ 1 or 2 (mod 4), the spectral sequence converges to

Σ n ⊕2 ℤ ℝP ℤ Λ0,nov/2F Λ0,nov . (5.14)

A close examination to the spectral sequence reveals that contributions to the direct 0 n n summand in (5.14) come from H (S ×S ; ℤ). Therefore (5.14) is isomorphic to

0 n n Σ n ℤ ℝP ℤ H (S ×S ; ℤ) ⊗ Λ0,nov/2F Λ0,nov .

Similarly, when n ≡ 0 or 3 (mod 4), the spectral sequence converges to

∗ n n ℤ H (S ×S ; ℤ) ⊗ Λ0,nov, CHAPTER 5. EXAMPLES AND COMPUTATIONS 55

as claimed in Proposition 5.12.

Corollary 5.15. Suppose n ≡ 1 or 2 (mod 4). Then we have

∗ n n ℤ ∼ 0 n n ℤ ℤ HF (S ×S ;Λnov) = H (S ×S ; ℤ) ⊗ (Λnov/2Λnov) ∼ ℤ ℤ ⊕2 = (Λnov/2Λnov) .

On the other hand, for n ≡ 0 or 3 (mod 4),

∗ n n ℤ ∼ ∗ n n ℤ HF (S ×S ;Λnov) = H (S ×S ; ℤ) ⊗ Λnov ∼ ℤ ⊕2 ℤ ⊕2 = (Λnov) ⊕ (Λnov) .

5.4 Morse-Bott Floer cohomology of the orientation covers of ℝPn

Now we investigate the orientation double covers of ℝPn and the associated La- grangian Floer theories. The manifold L in this case is the orientation double cover O(ℝPn) of ℝPn. When n is even, ℝPn in non-orientable and its orientation cover is the n n sphere double cover : S → ℝP . The ﬁber product L×L in this case is ΔSn ∐ΔSn as discussed in Section 5.3. Hence in this case, the analysis is the same as that in Section 5.3, and we have

Proposition 5.16. Suppose n is even and n ≡ 2 (mod 4). Then

∗ n n 0 n n Σ n ℤ ∼ ℤ ℝP ℤ HF (O(ℝP )×O(ℝP ); Λ0,nov) = H (S ×S ; ℤ) ⊗ Λ0,nov/2F Λ0,nov .

On the other hand, if n ≡ 0 (mod 4),

∗ n n ℤ ∼ ∗ n n ℤ HF (O(ℝP )×O(ℝP ); Λ0,nov) = H (S ×S ; ℤ) ⊗ Λ0,nov.

On the other hand, when n is odd, ℝPn is orientable. Its orientation cover is then the disjoint union of two copies of ℝPn with diﬀerent orientations. Let O(ℝPn) = L = CHAPTER 5. EXAMPLES AND COMPUTATIONS 56

n L0 ∐ L1, where each Lj is diﬀeomorphic to ℝP but with opposite orientations.

The ﬁber product L×L in this case is the disjoint union ΔL ∐ℛ. By deﬁnition,

ΔL is the union ΔL0 ∐ΔL1 of the diagonals of Lj’s. Clearly, the orientations on the two diagonals in ΔL are opposite to each others. On the other hand, ℛ is the set

{ (x, y) ∈ L×L ∣ x ∕= y, (x) = (y) } ,

where ∣Lj is just the identity. It then follows that ℛ is the union ℛ0 ∐ℛ1, where

ℛ0 = { (x, y) ∈ L0 ×L1 ∣ x ∕= y } ,

ℛ1 = { (x, y) ∈ L1 ×L0 ∣ x ∕= y } ,

with opposite orientations. We are going to show that:

Proposition 5.17. Suppose n = 2K + 1 is odd. When n ≡ 1 (mod 4), there is a unique non-zero diﬀerential

n n n 0 n n n+1 : H (O(ℝP )×O(ℝP ); ℤ) −→ H (O(ℝP )×O(ℝP ); ℤ) whose matrix is given by " # ±2I 0 , 0 ±2J where I is the identity, and " # 0 1 J = . 1 0 Hence in this case we have,

!⊕4 ∗ n n M Σ n ℤ ∼ ℤ ℝP ℤ HF (O(ℝP )×O(ℝP ); Λ0,nov) = Λ0,nov/2F Λ0,nov K+1 ∼ ∗ n n ℤ ⊕4 = HF (ℝP , ℝP ;Λ0,nov) , CHAPTER 5. EXAMPLES AND COMPUTATIONS 57

∗ n n ℤ where HF (ℝP , ℝP ;Λ0,nov) is the Morse-Bott Floer cohomology deﬁned in Fukaya- Oh-Ohta-Ono [FOOO06] .

On the other hand, when n ≡ 3 (mod 4), n+1 = 0. In this case,

∗ n n ℤ ∼ ∗ n ℤ ⊕4 HF (O(ℝP )×O(ℝP ); Λ0,nov) = H (ℝP ; ℤ) ⊗ Λ0,nov .

5.4.1 Floer coboundary operators and ℳ2( )

Before we prove Proposition 5.17, we need to discuss the coboundary operators and the associated moduli spaces. n When n is odd, the orientation double cover of ℝP is the disjoint union L0 ∐L1 of two oppositely oriented copies of ℝPn itself. Consider any pseudo-holomorphic disk

2 2 n n u :(D , ∂D ) → (ℂP , ℝP )

with two boundary marked points in the class j of the minimal Maslov index. Topo- logically, either it has a continuous lift entirely onto one of the components of ΔL with no jumps, or it has exactly two jumps with the two arcs of ∂D2 lifting to diﬀerent components of ΔL. Therefore, the relevant moduli spaces here have two non-empty components, one with no jumps and the other contains exactly two jumps. Denote these two compo- 0 2 nents by ℳ2( ) and ℳ2( ) respectively. Each of them consists of two components distinguished by the components of L and ℛ the lifts take place. Denote these com- 0,k 2,k ponents by ℳ2 ( ) and ℳ2 ( ) for k = 1, 2, and let their union be ℳ2( ). In terms of the evaluation maps, we have for k = 1, 2 that

0,k evlj : ℳ2 ( ) −→ ΔLk , 2,k evlj : ℳ2 ( ) −→ ℛk.

Similar to the discussions in Section 5.3, the only non-trivial contributions to the CHAPTER 5. EXAMPLES AND COMPUTATIONS 58

diﬀerential come from 0 and from

n n 0 0 j : CF (ΔL) ⊕ CF (ℛ) −→ CF (ΔL) ⊕ CF (ℛ), j = 1, 2,

n where j’s are the classes corresponding to the minimal Maslov index of ℝP . So again we only need to consider moduli spaces associated to the classes 1 and 2.

Proof of Proposition 5.17. We ﬁrst consider the case when there are no jumps. Since lifts in this case are continuous, evaluating at the two marked points gives a map

0,k evl := evl0×evl1 : ℳ2 ( j ) −→ (ΔLk ×ΔLk ) , whence we obtain the following lemma by similar arguments as in Lemma 5.14.

Lemma 5.18. Let evl := evl0×evl1 be the product of the evaluation maps. Then 0,k evl* ([ℳ2 ( j )]) gives the fundamental cycle [ΔLk ×ΔLk ] of ΔLk ×ΔLk . The diﬀerential in this case takes the form

n n 0 0 j : CF (ΔL0 ) ⊕ CF (ΔL1 ) −→ CF (ΔL0 ) ⊕ CF (ΔL1 ), j = 1, 2.

n Together with Lemma 5.18, it follows that for any Pk ∈ CF (ΔLk )

(n+1)/2 j Pk = ± [ΔLk ] e

0,k whose sign depends on the orientation of the moduli space ℳ2 ( j ). Similar to the discussions in Section 5.3, we have

⎧ (n+1)/2 ⎨±2 [ΔLk ] e when n ≡ 1 (mod 4), ( 1 + 2 )(Pk) = ⎩0 when n ≡ 3 (mod 4).

The case for two jump points is similar. In this case, the product of the evaluation maps at the two marked points becomes

2 evl := evl0×evl1 : ℳ2( j ) −→ (ℛ1 ×ℛ0) ∐ (ℛ0 ×ℛ1) , CHAPTER 5. EXAMPLES AND COMPUTATIONS 59

in which the component where the image lies in depends on the lifts of the two jump points. Again, similar arguments as in Lemma 5.14 lead to the following lemma.

2 Lemma 5.19. evl* ([ℳ2( j )]) gives the fundamental cycle [ℛ1 ×ℛ0] ⊕ [ℛ0 ×ℛ1] of

(ℛ1 ×ℛ0) ∐ (ℛ0 ×ℛ1). The diﬀerential in this case takes the form

n n 0 0 j : CF (ℛ0) ⊕ CF (ℛ1) −→ CF (ℛ0) ⊕ CF (ℛ1), j = 1, 2.

n Suppose P ∈ CF (ℛ0), then by Lemma 5.19,

(n+1)/2 j P = ± [ℛ1] e

2 whose sign depends on the orientation of the moduli space ℳ2( j ). Similarly, if n P ∈ CF (ℛ1), (n+1)/2 j P = ± [ℛ0] e .

n By the same arguments as in Section 5.3, it follows that for any P ∈ CF (ℛ0),

⎧ (n+1)/2 ⎨±2 [ℛ1] e when n ≡ 1 (mod 4), ( 1 + 2 )(P ) = ⎩0 when n ≡ 3 (mod 4);

n and for any P ∈ CF (ℛ1),

⎧ (n+1)/2 ⎨±2 [ℛ0] e when n ≡ 1 (mod 4), ( 1 + 2 )(P ) = ⎩0 when n ≡ 3 (mod 4).

Together they imply the ﬁrst half of Proposition 5.17. CHAPTER 5. EXAMPLES AND COMPUTATIONS 60

To ﬁnish the proof, consider the E0 page of the spectral sequence.

CF 0,2n+2 / ⋅ ⋅ ⋅

CF 0,n+1 / ⋅ ⋅ ⋅ / CF n,n+1

0 CF 0,0 / CF 1,0 / ⋅ ⋅ ⋅ / CF n,0

Let p,q p q/2 H := H (L×L; ℤ) ⊗ e .

We take homology of the horizontal arrows to obtain the E1 page.

H0,2n+2 ⋅ ⋅ ⋅ O n+1 H0,n+1 H1,n+1 0 ⋅ ⋅ ⋅ Hn,n+1 O n+1 H0,0 H1,0 0 H3,0 ⋅ ⋅ ⋅ Hn,0

From this, it follows that the spectral sequence degenerates on the E2 page. Therefore, when n ≡ 1 (mod 4), the spectral sequence converges to

!⊕4 M Σ n ℤ ℝP ℤ Λ0,nov/2F Λ0,nov . K+1

Similarly, when n ≡ 3 (mod 4), the spectral sequence converges to

∗ n ℤ ⊕4 H (ℝP ; ℤ) ⊗ Λ0,nov as claimed in Proposition 5.17. Chapter 6

Further directions

6.1 Obstructions to Lagrangian Floer cohomology

In general, moduli spaces do not always give rise to a differential for Floer cohomology. For Lagrangian covering spaces : L → (L), this may happen when the image of bounds disks of Maslov index 0. In this case, one cannot rule out disk bubbles and the square of the corresponding co-boundary map may not be zero. For an embedded Lagrangian (L), Fukaya at. el. have a way to get around this issue. One can first introduce more marked points and then cut down the dimension of the resulting moduli space by asking these marked point to pass through a certain co-chain b on (L), with some extra conditions. This leads to a deformation of the co-boundary operator, and the deformed one becomes a differential. It would be important to know if similar ideas can be applied in the case of clean self-intersections we are investigating.

6.2 Hamiltonian equivalence

Lagrangian Floer cohomology (over a certain class of Novikov rings), even when deﬁned, is not necessarily Hamiltonian isotopy invariant, not even for with embedded Lagrangians. This phenomenon is even more prominent for immersed Lagrangians. When only transverse double-point self-intersections are present, Akaho-Joyce [AJ08]

61 CHAPTER 6. FURTHER DIRECTIONS 62

observed one has to distinguish between the notions of local and global Hamiltonian equivalences. In the former one can slide sheets of L over each other and change the number of self-intersections, while the later one is induced by a global Hamiltonian isotopy and is thus more rigid. It turns out Floer cohomology (over a slightly enlarged Novikov ring) is invariant under global Hamiltonian equivalence. This fact essentially follows from the work of [FOOO06]. One may anticipate similar phenomena to happen in our case. ∗ n n ℤ If this were true, then, for example, the rank of HF (S ×S ;Λ0,nov) should give information on the lower bound of the number of intersections of the double cover : Sn → ℂPn, after a Hamiltonian perturbation such that the perturbed contains only transverse intersections. More precisely, by Proposition 5.12, when n ≡ 1 or 2 (mod 4), we have

0 n 0 Σ n Σ n ⊕2 ℤ ℝP ℤ ∼ ℤ ℝP ℤ H (S ; ℤ) ⊕ H (ℛ; ℤ) ⊗ Λ0,nov/2F Λ0,nov = Λ0,nov/2F Λ0,nov .

This will imply that the rank of H0(ℛ; ℤ) is 1. But ℛ is exactly the set of transverse intersections of the perturbed Lagrangian immersion : Sn → ℂPn, and contains only points. Hence the rank of H0(ℛ; ℤ) is 1 and the perturbed immersion must have at least 1 intersection. This will give another proof of the well-known fact by Gromov that Sn cannot be embedded in ℂPn as Lagrangian. Similarly, when n ≡ 0 or 3 (mod 4), we have

∗ n ∗ ℤ ∼ ℤ ⊕4 (H (S ; ℤ) ⊕ H (ℛ; ℤ)) ⊗ Λ0,nov = Λ0,nov .

This will imply that the rank of H∗(ℛ; ℤ) is 2. Hence the rank of H0(ℛ; ℤ) is 2 and the perturbed immersion must have at least 2 intersections. This will be something new. Appendix A

Spin and relatively spin structures

Deﬁnition A.1. Suppose En is an n-dimensional real vector bundle over a manifold k X. Then En has a spin structure if its stable bundle En ⊕ ℝ admits a trivialization k over the 1-skeleton of X which extends over the 2-skeleton,i where ℝ denotes a trivialized bundle. A spin structure is then deﬁned as a homotopy class of such trivializations.

It can be shown that the definition does not depend on k for k ⩾ 1. An oriented manifold X admits a spin structure if and only if its second Stiefel-Whitney class 1 w2(X) is trivial, and in such case, spin structures are parametrized by H (X; ℤ2) as a set. Recall that a framing of an n-dimensional vector bundle En is an ordered set of n everywhere linearly independent sections of En. Equivalently, a framing is a continuous map F : En → ℝn which is an isomorphism on each fiber. It can also be considered as a section of the associated principal bundle of En. Therefore, a bundle En admits a framing if and only if it is trivial, and a framing determines and is determined by a trivialization of En. Consider a trivialized (n + k)-dimensional real vector bundle En+k over X, with a k-dimensional trivialized subbundle Ek. It is then natural to ask to what extent a trivialization on Ek affects the trivialization of the orthogonal n-dimensional bundle.

iNote that if a trivialization of the 1-skeleton extends to the 2-skeleton, it extends in a homo- topically unique way.

63 APPENDIX A. SPIN AND RELATIVELY SPIN STRUCTURES 64

Fix a framing F of En+k as a reference. Up to homotopy, we may assume all framings are orthonormal. The trivializations of En+k and Ek induce framings on the respective bundles, and we have the following diagram of maps

O(n + k) 8 sn+k qq qqq p qqq qqq q / O(n + k)/O(n) X sk where p is the projection of the ﬁbration. The quotient O(n + k)/O(n) is the Stiefel n+k n+k manifold Vk(ℝ ) of all k-dimensional frames in ℝ . This space is (n − 1)– connected with the ﬁrst non-vanishing homotopy group being n. Therefore, on the n-skeleton of X, the maps p ∘ sn+k and sk are homotopic. The homotopy can ′ ′ be covered by a homotopy of sn+k to a map s such that p ∘ s = sk. This induces a k framing sn, and thus a trivialization, on the n-dimensional bundle orthogonal to E over the n-skeleton of X.

The sum sk⊕sn of the framings is homotopic to the framing induced by the original trivialization of E. Notice that sk ⊕ sn is the composition of a map X → O(k) with the inclusion O(k) ,→ O(n + k). Since the later inclusion induces an injection on the homotopy groups j for j < n−1, the framing sn, and thus the induced trivialization, is unique up to homotopy on the (n − 1)-skeleton of X. The upshot is that a trivialized k-dimensional subbundle of a trivialized (n + k)- dimensional bundle En+k determines a trivialization of the orthogonal n-dimensional subbundle up to homotopy over the (n−1)-skeleton of the base manifold X. Therefore k the trivializations of En ⊕ ℝ naturally correspond to trivializations of En over the 2-skeleton if n ⩾ 3. So in practice, to apply deﬁnition A.1, one does not need to 2 stabilize for n ⩾ 3. When n = 2, we can add ℝ, and when n = 1, we add ℝ . Recall the deﬁnition of spin groups.

Deﬁnition A.2. The spin group Spin(n) in dimension n is the double cover of SO(n), non-trivial for n ⩾ 2. APPENDIX A. SPIN AND RELATIVELY SPIN STRUCTURES 65

In particular, the spin group gives rise to the short exact sequence

0 −→ ℤ/2ℤ −→ Spin(n) −−→ SO(n) −→ 1.

Denote by PSO(n)(E) the principal SO(n) bundle of E. There an equivalent deﬁ- nition of the spin structure of E.

Deﬁnition A.3. An oriented real bundle E admits a spin structure if there exists a principal Spin(n) bundle PSpin(n)(E) together with a 2-fold covering map

PSpin(n)(E) −−→ PSO(n)(E) such that the following diagram commutes:

PSpin(E) ×Spin(n) / PSpin(E) GG GG GG GG G# × X w; ww ww ww ww PSO(E) ×SO(n) / PSO(E)

This happens if and only if the second Stiefel-Whitney class w2(E) of E is zero. References

[AJ08] Manabu Akaho and Dominic Joyce, Immersed Lagrangian Floer Theory, ArXiv e-prints (2008).

[APS75] Michael F. Atiyah, V. K. Patodi, and Isadore M. Singer, Spectral asym- metry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.

[BW97] Sean Bates and Alan Weinstein, Lectures on the geometry of quantiza- tion, Berkeley Mathematics Lecture Notes, vol. 8, American Mathemat- ical Society, Providence, RI, 1997.

[CL06] Octav Cornea and Fran¸coisLalonde, Cluster homology: an overview of the construction and results, Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 1–12 (electronic).

[Flo88a] Andreas Floer, Morse theory for Lagrangian intersections, J. Diﬀerential Geom. 28 (1988), no. 3, 513–547.

[Flo88b] , A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 4, 393–407.

[Flo88c] , The unregularized gradient ﬂow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813.

[Flo89] , Witten’s complex and inﬁnite-dimensional Morse theory, J. Dif- ferential Geom. 30 (1989), no. 1, 207–221.

66 REFERENCES 67

[FO97] Kenji Fukaya and Yong-Geun Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997), no. 1, 96–180.

[FOOO06] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, La- grangian intersection Floer theory – anomaly and obstruction, 2006, http://www.math.kyoto-u.ac.jp/˜fukaya/, preprint.

[FOOO09a] , Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.

[FOOO09b] , Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.

[Fuk02] Kenji Fukaya, Floer homology and mirror symmetry. II, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 31–127.

[Gro85] Mikhael Gromov, Pseudoholomorphic curves in symplectic manifolds, In- vent. Math. 82 (1985), no. 2, 307–347.

[Liu02] Chiu-Chu Melissa Liu, Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $Sˆ1$- Equivariant Pair, ArXiv Mathematics e-prints (2002).

[LM85] Robert B. Lockhart and Robert C. McOwen, Elliptic diﬀerential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409–447.

[MS04] Dusa McDuﬀ and Dietmar Salamon, J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. REFERENCES 68

[Oh93a] Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993.

[Oh93b] , Floer cohomology of Lagrangian intersections and pseudo- holomorphic disks. II. (CPn, RPn), Comm. Pure Appl. Math. 46 (1993), no. 7, 995–1012.

[Oh95] , Riemann-Hilbert problem and application to the perturbation theory of analytic discs, Kyungpook Math. J. 35 (1995), no. 1, 39–75.

[Oh96] , Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices (1996), no. 7, 305– 346.

[Po´z99] Marcin Po´zniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 119– 181.

[RS93] Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topol- ogy 32 (1993), no. 4, 827–844.

[RS95] , The spectral ﬂow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33.

[RT95] Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Diﬀerential Geom. 42 (1995), no. 2, 259–367.

[Sei08] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z¨urich, 2008.

[Sol06] Jake P. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, ArXiv Mathematics e-prints (2006).